ms. parsons math class - background: prehistoric rock art...

353

About the Chapter Project

The heating and cooling of objects can bemodeled by functions. Throughout this chapterand in the Chapter Project, Warm Ups, you willmodel the heating and cooling of a temperatureprobe over several temperature ranges in orderto find an appropriate general model for thesephenomena.

After completing the Chapter Project, you will beable to do the following:

● Collect real-world data on the heating andcooling of an object, and determine anappropriate exponential function to modelthe heating and cooling of an object.

● Make predictions about the temperature of anobject that is heating or cooling to a constantsurrounding temperature.

● Verify Newton’s law of cooling.

In the Portfolio Activities for Lessons 6.1 and 6.4and in the Chapter Project, you will need to use a program like the one shown on thecalculator screen at right to collect temperaturedata with a CBL.

About the Portfolio Activities

Throughout the chapter, you will be givenopportunities to complete Portfolio Activitiesthat are designed to support your work on theChapter Project.

● Using a CBL to collect cooling temperaturedata in a laboratory setting is included in thePortfolio Activity on page 361.

● Comparing different models for the coolingtemperature data is included in the PortfolioActivity on page 369.

● Using a CBL to collect warming temperaturedata and performing appropriatetransformations on regression equations areincluded in the Portfolio Activity on page 384.

● Comparing Newton’s law of cooling withregression models from empirical data isincluded in the Portfolio Activity on page 409.

Background: Prehistoric rock artfrom the Canyon de ChellyNational Monument, Arizona;

Right: Anasazi sandal, 700–900years old, found at NavajoNational Monument, Arizona

354 CHAPTER 6

ObjectivesObjectives

● Determine the multiplierfor exponential growthand decay.

● Write and evaluateexponentialexpressions to modelgrowth and decaysituations.

Exponential Growthand Decay

Bacteria are very small single-celled organisms that live almost everywhere onEarth. Most bacteria are not harmful to humans, and some are helpful, such asthe bacteria in yogurt.

Bacteria reproduce, or grow in number, by dividing. The total number ofbacteria at a given time is referred to as the population of bacteria. When eachbacterium in a population of bacteria divides, the population doubles.

Modeling Bacterial Growth

You will need: a calculator

You can use a calculator to model thegrowth of 25 bacteria, assuming that theentire population doubles every hour.

First enter 25. Then multiply this number by 2 to find the population of bacteria after1 hour. Repeat this doubling procedure to find the population after 2 hours.

A P P L I C A T I O N

BIOLOGY

TECHNOLOGYGRAPHICS

CALCULATOR

Keystroke Guide, page 418

Exponential growth anddecay can be used to model anumber of real-world situations,such as population growth ofbacteria and the elimination ofmedicine from the bloodstream.

WhyWhy

1. Copy and complete the table below.

2. Write an algebraic expression that represents the population of bacteriaafter n hours. (Hint: Factor out 25 from each population figure.)

3. Use your algebraic expression to find the population of bacteria after 10 hours and after 20 hours.

4. Suppose that the initial population of bacteria was 75 instead of 25.Find the population after 10 hours and after 20 hours.

You can represent the growth of an initial population of 100 bacteria thatdoubles every hour by creating a table.

The bar chart at right illustrateshow the doubling pattern ofgrowth quickly leads to largenumbers.

Assuming an initial population of 100 bacteria, predict the population ofbacteria after 5 hours and after 6 hours.

The population after n hours can be represented by the following exponentialexpression:

n times

100 × 2 × 2 × 2 × . . . × 2 = 100 × 2n

This expression, 100 • 2n, is called an exponential expression because theexponent, n, is a variable and the base, 2, is a fixed number. The base of anexponential expression is commonly referred to as the multiplier.

355LESSON 6.1 EXPONENTIAL GROWTH AND DECAY

0 1 2 3 4 . . . n

100 200 400 800 1600 . . . 100(2)nTime (hr)

Population

+1 +1 +1 +1

× 2 × 2 × 2 × 2

C O N N E C T I O N

PATTERNS IN DATA

CHECKPOINT ✔

�

Time (hr)

Pop

ula

tion

200

400

600

800

1000

1200

1400

1600

0 1 2 3 4

CHECKPOINT ✔

Time (hr) 0 1 2 3 4 5 6

Population 25 50 100

356 CHAPTER 6

The population of the United States was 248,718,301 in 1990 and wasprojected to grow at a rate of about 8% per decade. [Source: U.S. CensusBureau]

Predict the population, to the nearest hundred thousand, for the years 2010and 2025.

SOLUTION

1. To obtain the multiplier for exponential growth, add the growth rate to 100%.

100% + 8% = 108%, or 1.08

2. Write the expression for the population n decades after 1990.

248,718,301 • (1.08)n

3. Since the year 2010 is 2 decades after 1990, substitute 2 for n.

248,718,301(1.08)n

= 248,718,301(1.08)2

= 290,105,026.3

To the nearest hundred thousand,the predicted population for 2010 is 290,100,000.

These predictions are based on the assumption that the growth rate remains a constant 8% per decade.

The population of Brazil was about 162,661,000 in 1996 and was projected to grow at a rate of about 7.7% per decade. Predict the population, to thenearest hundred thousand, of Brazil for 2016 and 2020. [Source: U.S. CensusBureau]

If a population’s growth rate is 1% per year, what is the population’s growthrate per decade?

E X A M P L E 1

Since the year 2025 is 3.5 decades after 1990, substitute 3.5 for n.

248,718,301(1.08)n

= 248,718,301(1.08)3.5

= 325,604,866

To the nearest hundred thousand,the predicted population for 2025 is 325,600,000.

TRY THIS

CRITICAL THINKING

Modeling Human Population Growth

Human populations grow much more slowly than bacterialpopulations. Bacterial populationsthat double each hour have agrowth rate of 100% per hour.The population of the UnitedStates in 1990 was growing at arate of about 8% per decade.

In Example 1, you will use thisgrowth rate to make predictions.


DEMOGRAPHICS

TECHNOLOGYSCIENTIFIC

CALCULATOR


The rate at which caffeine is eliminated from the bloodstream ofan adult is about 15% per hour. An adult drinks a caffeinated soda, and the caffeine in his or her bloodstream reaches a peak level of 30 milligrams.

Predict the amount, to the nearest tenth of a milligram, of caffeine remaining 1 hour after the peak level and 4 hours after the peak level.

SOLUTION

1. To obtain the multiplier for exponential decay, subtract the rate of decay from 100%. The multiplier is found as follows:

100% − 15% = 85%, or 0.85

2. Write the expression for the caffeine level x hours after the peak level.

30(0.85)x

3. Substitute 1 for x.

30(0.85)x

= 30(0.85)1

= 25.5

The amount of caffeine remaining 1 hour after the peak level is 25.5 milligrams.

A vitamin is eliminated from the bloodstream at a rate of about 20% per hour. The vitamin reaches a peak level in the bloodstream of 300milligrams. Predict the amount, to the nearest tenth of a milligram, of thevitamin remaining 2 hours after the peak level and 7 hours after the peak level.


HEALTH

E X A M P L E 2

Substitute 4 for x.

30(0.85)x

= 30(0.85)4

≈ 15.7

The amount of caffeine remaining 4 hours after the peak level is about15.7 milligrams.

TRY THIS

Modeling Biological Decay

Caffeine is eliminated from thebloodstream of a child at a rate ofabout 25% per hour. This exponentialdecrease in caffeine in a child’sbloodstream is shown in the bar chart.

A rate of decay can be thought of as a negative growth rate. To obtain themultiplier for the decrease in caffeine in the bloodstream of a child, subtractthe rate of decay from 100%. Thus, the multiplier is 0.75, as calculated below.

100% − 25% = 75%, or 0.75

Time (hours)

20%

40%

60%

80%

100%

0 1 2 3 4 5

Caffeine Elimination in Children

Ori

gin

al c

affe

ine

rem

ain

ing


CALCULATOR

Caffeine is an ingredient in coffee, tea,chocolate, and some soft drinks.

CHAPTER 6

Find the multiplier for each rate of exponential growth or decay.

(EXAMPLES 1 AND 2)

5. 5.5% growth 6. 0.25% growth 7. 3% decay 8. 0.5% decay

Evaluate each expression for x = 3. (EXAMPLES 1 AND 2)

9. 2x 10. 50(3)x 11. 0.8x 12. 100(0.75)x

13. DEMOGRAPHICS The population of Tokyo-Yokohama, Japan, was about28,447,000 in 1995 and was projected to grow at an annual rate of 1.1%.Predict the population, to the nearest hundred thousand, for the year2004. [Source: U.S. Census Bureau] (EXAMPLE 1)

14. HEALTH A certain medication is eliminated from the bloodstream at a rate of about 12% per hour. The medication reaches a peak level in thebloodstream of 40 milligrams. Predict the amount, to the nearest tenth ofa milligram, of the medication remaining 2 hours after the peak level and3 hours after the peak level. (EXAMPLE 2)

Find the multiplier for each rate of exponential growth or decay.

15. 7% growth 16. 9% growth 17. 6% decay

18. 2% decay 19. 6.5% growth 20. 8.2% decay

21. 0.05% decay 22. 0.08% growth 23. 0.075% growth

Given x = 5, y = �35

�, and z = 3.3, evaluate each expression.

24. 2x 25. 3y 26. 22x

27. 50(2)3x 28. 25(2)z 29. 25(2)y

30. 100(3) x�1 31. 10(2)z�2 32. 22y�1

33. 100(2)4z 34. 100(0.5)3z 35. 75(0.5)2y

Guided Skills Practice

A P P L I C A T I O N S

Practice and Apply

ExercisesExercises

Communicate

1. What type of values of n are possible in the bacterial growth expression 25 • 2n and in the United States population growth expression 248,718,301 • (1.08)n?

2. Explain how the United States population growth expression 248,718,301 • (1.08)n incorporates the growth rate of 8% per decade.

3. What assumption(s) do you make about a population’s growth when youmake predictions by using an exponential expression?

4. Describe the difference between the procedures for finding the multiplierfor a growth rate of 5% and for a decay rate of 5%.

358

HomeworkHelp OnlineGo To: go.hrw.comKeyword:MB1 Homework Help

for Exercises 24–35

Predict the population of bacteria for each situation and time period.

36. 55 bacteria that double every houra. after 3 hours b. after 5 hours

37. 125 bacteria that double every houra. after 6 hours b. after 8 hours

38. 33 E. coli bacteria that double every 30 minutesa. after 1 hour b. after 6 hours

39. 75 E. coli bacteria that double every 30 minutesa. after 2 hours b. after 3 hours

40. 225 bacteria that triple every houra. after 1 hour b. after 3 hours

41. 775 bacteria that triple every houra. after 2 hours b. after 4 hours

42. Suppose that you put $2500 into a retirement account that grows with aninterest rate of 5.25% compounded once each year. After how many yearswill the balance of the account be at least $15,000?

PATTERNS IN DATA Determine whether each table represents a linear,

quadratic, or exponential relationship between x and y.

43. 44. 45. 46.

47. DEMOGRAPHICS Thepopulation of Indonesiawas 191,256,000 in 1990and was growing at arate of 1.9% per year.Predict the population,to the nearest hundredthousand, of Indonesiain 2010. [Source: U.S.Census Bureau]

48. HEALTH A dye isinjected into thepancreas during acertain medicalprocedure. A physician injects 0.3 grams ofthe dye, and a healthy pancreas will secrete 4% of the dye each minute. Predict the amount of dye remaining, to the nearesthundredth of a gram, in a healthy pancreas 30 minutes after the injection.

49. DEMOGRAPHICS The population of China was 1,210,005,000 in 1996 andwas growing at a rate of about 6% per decade. Predict the population, tothe nearest hundred thousand, of China in 2016 and in 2021. [Source: U.S. Census Bureau]


C H A L L E N G E

x

0

1

2

3

y

2

4

8

16

x

1

2

3

4

y

1

3

9

27

x

0

2

4

6

y

6

10

14

18

x

0

3

6

9

y

−2

7

34

79

C O N N E C T I O N


Bali, Indonesia

50. PHYSICAL SCIENCE Suppose that a camera filter transmits 90% of the lightstriking it, as illustrated at left.a. If a second filter of the same type is added, what portion of light is

transmitted through the combination of the two filters?b. Write an expression to model the portion of light that is transmitted

through n filters.c. Calculate the portion of light transmitted through 4, 5, and 6 filters.

51. DEMOGRAPHICS The population of India was 952,108,000 in 1996 and wasgrowing at a rate of about 1.3% per year. [Source: U.S. Census Bureau]a. Predict the population, to the nearest hundred thousand, of India in

2000 and in 2010.b. Find the growth rate per decade that corresponds to the growth rate of

1.3% per year.c. Suppose that the population growth rate of India slows to 1% per year

after the year 2000. What is the predicted population, to the nearesthundred thousand, of India in 2010?

52. CHEMISTRY A dilution is commonly used to obtain thedesired concentration of a sample. For example,suppose that 1 milliliter of hydrochloric acid, or HCl,is combined with 9 milliliters of a buffer. The concentration of the resulting mixture is �1

10�

of the original concentration of HCl.a. Suppose that this dilution is performed again

with 1 millimeter of the already dilutedmixture and 9 milliliters of buffer. What is theconcentration of the resulting mixture(compared with the original concentration)?

b. Write an expression to model theconcentration of HCl in the resultingmixture after repeated dilutions asdescribed in part a.

c. What is the concentration of theresulting mixture (compared tothe original concentration) after5 repeated dilutions?

53. SPACE SCIENCE The first stage of theSaturn 5 rocket that propelledastronauts to the moon burnedabout 8% of its remaining fuelevery 15 seconds and carriedabout 600,000 gallons offuel at liftoff. Estimatethe amount of fuelremaining, to thenearest tenthousand gallons,in the first stage 2 minutes after liftoff.

360 CHAPTER 6

?%

?%

90%

100%

Filter 1

Filter 2

Filter 3


A C T I V

I TY

PO

RTFOLIO

Refer to the discussions of the PortfolioActivities and Chapter Project on page 353 forbackground on this activity.

You will need a CBL with a temperature probe,a glass of ice water, and a graphics calculator.

1. First use the CBL to find the temperature ofthe air. Then place the probe in the icewater for 2 minutes. Record 30 CBLreadings taken at 2-second intervals. Take afinal reading at the end of the 2 minutes.

2. a. Use the linear regression feature on yourcalculator to find a linear function thatmodels your first 30 readings. (Use thevariable t for the time in seconds).

b. Use your linear function to predict the temperature of the probe after 2 minutes, or 120 seconds. Compare this

prediction with your actual 2-minutereading.

c. Discuss the usefulness of your linearfunction for modeling the coolingprocess. (You may want to illustrate youranswer with graphs.)

Save your data and results for use in theremaining Portfolio Activities.

WORKING ON THE CHAPTER PROJECTYou should now be able to complete Activity 1of the Chapter Project.

Look Beyond


Evaluate each expression. (LESSON 2.2)

54. 4−2 55. (�12�)

−156. 25

�32�

57. 49�12�

Simplify each expression, assuming that no variable equals zero. Write

your answer with positive exponents only. (LESSON 2.2)

58. ��2xx−2

3��2

59. ��mn

−

−

1

3n2

��−3

60. ��−2aa

3

2bb

−

−

2

3��−1

61. �(2

3

y

x

2

y

y−

)4

−2

�

Identify each transformation from the graph of f (x) = x2 to the graph

of g. (LESSON 2.7)

62. g(x) = 6x2 63. g(x) = (−2x)2 64. g(x) = − �12�x2 + 1

65. g(x) = −(0.5x)2 + 3 66. g(x) = (x − 3)2 + 2 67. g(x) = −5(x − 2)2 − 4

State whether each parabola opens up or down and whether the

y-coordinate of the vertex is the maximum or minimum value of the

function. (LESSON 5.1)

68. f(x) = �12�x2 69. f(x) = −2x2 − x + 1 70. f(x) = 3 − 5x − x2

71. INVESTMENTS Suppose that you want to invest $100 in a bank account thatearns 5% interest compounded once at the end of each year. Determine thebalance after 10 years.


Look Back

362 CHAPTER 6

Exponential Functions

Objectives

● Classify an exponential functionas representingexponential growth orexponential decay.

● Calculate the growthof investments undervarious conditions.

x y = 2x

−3 2−3 =

−2 2−2 =

−1 2−1 =0 20 = 11 21 = 2

�2� 2 ≈ 2.67

2 22 = 43 23 = 8

�2�

1�8

1�4

1�2

You can use exponentialfunctions to calculate the valueof investments that earncompound interest and tocompare different investmentsby calculating effective yields.

WhyWhy

Consider the function y = x2 and y = 2x. Both functions have a base and anexponent. However, y = x2 is a quadratic function, and y = 2x is an exponentialfunction. In an exponential function, the base is fixed and the exponent isvariable.

Examine the table at left andthe graph at right of theexponential function y = 2x.

Notice that as x-values decrease,the y-values for y = 2x get closerand closer to 0, approaching the x-axis as an asymptote. Anasymptote is a line that a graphapproaches (but does notreach) as its x- or y-valuesbecome very large or very small.

Exponential Function

The function f(x) = bx is an exponential function with base b, where b isa positive real number other than 1 and x is any real number.

EXPONENT

BASE

31

1

2

3

4

5

6

7

8

−1−2−3 2x

√2

2√2

y

y = 2x

The graphof y = 2 x

approachesthe x-axisbut neverreaches it.

Notice thatthe domainof y = 2 x

includesirrationalnumbers,such as �2�.

363LESSON 6.2 EXPONENTIAL FUNCTIONS

Investigating Exponential Functions

You will need: a graphics calculator

1. Graph y1 = 3x, y2 = 2x, and y3 = (1.5)x onthe same screen.

2. For what value of x is y1 = y2 = y3 true?For what values of x is y1 > y2 > y3 true? For what values of x is y1 < y2 < y3 true?

3. Graph y4 = (�13�)

x, y5 = (�

12�)

x, and y6 = (�1

1.5�)

xon the same screen as y1,

y2, and y3.

4. Look for a pattern. Examine each corresponding pair of functions.

y1 = 3 and y4 = (�13�)

xy2 = 2x and y5 = (�

12�)

x

y3 = (1.5)x and y6 = (�11.5�)

x

How are the graphs of each corresponding pair of functions related?How are the bases of each corresponding pair of functions related?

5. For what values of b does the graph of y = bx rise from left to right? For what values of b does the graph of y = bx fall from left to right?

The graphs of f(x) = 2x and g(x) = (�12�)

xexhibit the two typical behaviors for

exponential functions.

Recall from Lesson 2.7 that the graphs of f and g are reflections of one another

across the y-axis because g(x) = f(�x) = 2−x = (�12�)

x.

Exponential Growth and Decay

When b > 1, the function f(x) = bx represents exponential growth.

When 0 < b < 1, the function f(x) = bx represents exponential decay.

CHECKPOINT ✔

C O N N E C T I O NTRANSFORMATIONS

3

4

5

y

x1

g(x) = ( ) f(x) = 2x

2

2

1

−2 −1

exponentialgrowth

exponential decay

y-intercept

12

x

TECHNOLOGYGRAPHICS

CALCULATOR


PROBLEM SOLVING

g(x) = (�12�) x

is a decreasingexponential functionbecause its base is apositive number less than 1.

f(x) = 2 x is an increasingexponential functionbecause its base is a positive numbergreater than 1.

Exponential growth functions and exponential decay functions of the form y = bx have the same domain, range, and y-intercept. For example:

Recall from Lesson 2.7 that y = a • f(x) represents a vertical stretch orcompression of the graph of y = f(x). This transformation is applied toexponential functions in Example 1.

Graph f(x) = 2x along with each function below. Tell whether each functionrepresents exponential growth or exponential decay. Then give the y-intercept.

a. y = 3 • f(x) b. y = 5 • f(−x)

SOLUTION

a. y = 3 • f(x) = 3 • 2x

The function y = 3 • 2x representsexponential growth because the base, 2, is greater than 1.

The y-intercept is 3 because thegraph of f(x) = 2x, which has a y-intercept of 1, is stretched by a factor of 3.

b. y = 5 • f(−x) = 5 • 2−x = 5 • (�12�)

x

The function y = 5 • (�12�)

xrepresents

exponential decay because the base,

�12�, is less than 1.

The y-intercept is 5 because the graph of f(x) = 2x, which has a y-intercept of 1, is stretched by a factor of 5.

Graph f(x) = 2x along with each function below. Tell whether each functionrepresents exponential growth or exponential decay. Then give the y-intercept.

a. y = �13� • f(x) b. y = �

14� • f(−x)

What transformation of f occurs when a < 0 in y = a • f(x)?

Describe the effect on the graph of f(x) = bx when b > 1 and b increases.Describe the effect on the graph of f(x) = bx when 0 < b < 1 and b decreases.

364 CHAPTER 6

E X A M P L E 1

C O N N E C T I O N

TRANSFORMATIONS

TRY THIS

CRITICAL THINKING

CHECKPOINT ✔

Function Domain Range y-intercept

f(x) = 2x all real all positive1

numbers real numbers

g(x) = (�12�)

x all real all positive1

numbers real numbers

TECHNOLOGYGRAPHICS

CALCULATOR


The y-intercept of y = 3 • 2 x is 3.

The y-intercept of y = 2 x is 1.

The y-intercept of y = 5 • (�

12�)

xis 5.

The y-intercept of y = 2 x is 1.


E X A M P L E 2

Compound Interest

The growth in the value of investments earning compound interest is modeledby an exponential function.

Find the final amount of a $100investment after 10 years at 5%interest compounded annually,quarterly, and daily.

SOLUTION

In this situation, the principal is $100,the annual interest rate is 5%, and thetime period is 10 years. Thus, P = 100,r = 0.05, and t = 10. The table showscalculations for n = 1, n = 4, and n = 365.

Describe what happens to the final amount as the number of compoundingperiods increases.

Effective Yield

Suppose that you buy an item for $100 and sell the item one year later for$105. In this case, the effective yield of your investment is 5%. The effectiveyield is the annually compounded interest rate that yields the final amount ofan investment. You can determine the effective yield by fitting an exponentialregression equation to two points.

Compound Interest Formula

The total amount of an investment, A, earning compound interest is

A(t) = P(1 + �nr

�)nt

,

where P is the principal, r is the annual interest rate, n is the number oftimes interest is compounded per year, and t is the time in years.


INVESTMENTS

CHECKPOINT ✔


INVESTMENTS

Compounding n Finalperiod amount

annually 1 A(10) = 100(1 + �0.105�)

1 • 10

$162.89

quarterly 4 A(10) = 100(1 + �0.405�)

4 • 10

$164.36

daily 365 A(10) = 100(1 + �03.6055�)

365 • 10

$164.87

A(10) = 100(1 + �0.

n05�)

n • 10TECHNOLOGYSCIENTIFIC

CALCULATOR

ExercisesExercises

366 CHAPTER 6

A collector buys a painting for $100,000 at thebeginning of 1995 and sells it for $150,000 at the beginning of 2000.Use an exponential regression equation to findthe effective yield.

SOLUTION

1. Find the exponential equation that represents this situation.

To find effective yield, the interest iscompounded annually, so n = 1.

From 1995 to 2000 is 5 years, so t = 5.

A(t) = P(1 + �nr

�)nt

150,000 = 100,000(1 + �1r

�)1 • 5

150,000 = 100,000(1 + r)5

2. Enter the two points that represent the given information, (0, 100,000)and (5, 150,000). Find and graphthe exponential regressionequation that fits the points.

3. The multiplier is about 1.084,so the effective yield, is about 1.084 − 1 = 0.084, or 8.4%.

Find the effective yield for a painting bought for $100,000 at the end of 1994and sold for $200,000 at the end of 2004.

1. If b > 0 and the graph of y = bx falls from left to right, describe thepossible values of b.

2. Compare the domain and range of y = 3x with the domain and range

of y = (�13�)

x.

3. Describe how the y-intercept of the graph of f(x) = 2(5)x is related to thevalue of a in f(x) = ab x.

4. How are the functions y = x2 and y = 2x similar, and how are they different?

TRY THIS

Communicate

E X A M P L E 3

C O N N E C T I O N

STATISTICS

TECHNOLOGYGRAPHICS

CALCULATOR


(0, 100,000)

The exponential regressionequation is y ≈ 100,000(1.084) x.

(5, 150,000)

Activities OnlineGo To: go.hrw.comKeyword:MB1 Medicine


Tell whether each function represents exponential growth or

exponential decay, and give the y-intercept. (EXAMPLE 1)

5. f(x) = (�12�)

x6. g(x) = 3(2)x 7. k(x) = 5(0.5)x

8. INVESTMENTS Find the final amount of a $250 investment after 5 years at6% interest compounded annually, quarterly, and daily. (EXAMPLE 2)

9. INVESTMENTS Find the effective yield for a $2000 investment that is worth$4000 after 15 years. (EXAMPLE 3)

Identify each function as linear, quadratic, or exponential.

10. g(x) = 10x + 3 11. k(x) = (77 − x)x . 12. f(x) = 12(2.5)x

13. k(x) = 0.5x − 3.5 14. g(x) = (2200)3.5x 15. h(x) = 0.5x2 + 7.5

Tell whether each function represents exponential growth or decay.

16. y(x) = 12(2.5)x 17. k(x) = 500(1.5)x 18. y(t) = 45(�14�)

t

19. d(x) = 0.125(�12�)

x20. g(x) = 0.25(0.8)x 21. s(k) = 0.5(0.5)k

22. m(x) = 222(0.9)x 23. f(k) = 722−k 24. g(x) = 0.5(787)−x

Match each function with its graph.

25. y = 2x 26. y = 2(3)x

27. y = 2(�13�)

x28. y = (�

12�)

x

Find the final amount for each investment.

29. $1000 at 6% interest compounded annually for 20 years

30. $1000 at 6% interest compounded semiannually for 20 years

31. $750 at 10% interest compounded quarterly for 10 years

32. $750 at 5% interest compounded quarterly for 10 years

33. $1800 at 5.65% interest compounded daily for 3 years

34. $1800 at 5.65% interest compounded daily for 6 years

35. Graph f(x) = 2x, g(x) = 5x, and h(x) = 8x.a. Which function exhibits the fastest growth? the slowest growth?b. What is the y-intercept of each function?c. State the domain and range of each function.

36. Graph a(x) = (�12�)x

, b(x) = (�15�)x

, and c(x) = (�18�)x

.

a. Which function exhibits the fastest decay? the slowest decay?b. What is the y-intercept of each function?c. State the domain and range of each function.

37. Describe when the graph of f(x) = abx is a horizontal line.

y

x1

a b c d

2 3

2

−2−3 −1

3

4

1

C H A L L E N G E



Practice and Apply

C O N N E C T I O N S

368 CHAPTER 6

TRANSFORMATIONS Graph each pair of functions and describe the

transformations from f to g.

38. f(x) = (�12�)

xand g(x) = 5(�

12�)

x39. f(x) = (�1

10�)

xand g(x) = 0.5(�1

10�)

x

40. f(x) = 2x and g(x) = 3(2)x + 1 41. f(x) = 10x and g(x) = 2(10)x − 3

42. f(x) = 10x and g(x) = 3(10)x + 2 43. f(x) = 2x and g(x) = 5(2)x − 1

44. f(x) = 3(�12�)

xand g(x) = 3(2x) 45. f(x) = (�

13�)

xand g(x) = 2(3)−x

46. TRANSFORMATIONS Describe how each transformation of f(x) = bx affectsthe domain and range, the asymptotes, and the intercepts.a. a vertical stretch b. a vertical compressionc. a horizontal translation d. a vertical translatione. a reflection across the y-axis

STATISTICS Use an exponential regression equation to find the effective

yield for each investment. Assume that interest is compounded only

once each year.

47 a $1000 mutual fund investment made at the beginning of 1990 that isworth $1450 at the beginning of 2000

48 a house that is bought for $75,000 at the end of 1995 and that is worth$95,000 at the end of 2005

STATISTICS Use an exponential regression equation to model the annual

rate of inflation, or percent increase in price, for each item described.

49 a half-gallon of milk cost $1.37 in 1989 and $1.48 in 1995 [Source: U.S. Bureau of Labor Statistics]

50 a gallon of regular unleaded gasoline cost $0.93 in 1986 and $1.11 in 1993[Source: U.S. Bureau of Labor Statistics]

51. INVESTMENTS Find the finalamount of a $2000 certificate ofdeposit (CD) after 5 years at anannual interest rate of 5.51%compounded annually.

52. INVESTMENTS Consider a $1000investment that is compoundedannually at three different interestrates: 5%, 5.5%, and 6%.a. Write and graph a function for

each interest rate over a time period from 0 to 60 years.b. Compare the graphs of the three functions.c. Compare the shapes of the graphs for the first 10 years with the shapes

of the graphs between 50 and 60 years.

53. INVESTMENTS The final amount for $5000 invested for 25 years at 10%annual interest compounded semiannually is $57,337.a. What is the effect of doubling the amount invested?b. What is the effect of doubling the annual interest rate?c. What is the effect of doubling the investment period?d. Which of the above has the greatest effect on the final amount of

the investment?

A P P L I C A T I O N SCertificate of Deposit



A C T I V

I TY

PO

RTFOLIO

For this activity, use the data collected in thePortfolio Activity on page 361.

1. a. Use the quadratic regression feature onyour calculator to find a quadraticfunction that models your first 30readings.

b. Use your quadratic function to predictthe temperature of the probe after 2minutes. Compare this prediction withyour actual 2-minute reading.

c. Discuss the usefulness of your quadratic function for modeling the cooling process.

2. Now use the exponential regression featureon your calculator to find an exponentialfunction that models your first 30 readings,and repeat parts b and c of Step 1.

Save your data and results to use in theremaining Portfolio Activities.



Find the inverse of each function. State whether the inverse is a

function. (LESSON 2.5)

54. {(−2, 4), (−3, −1), (2, 2), (3, 4)}

55. {(7, 2), (3, −1), (2, 2), (0, 0)}

56. y = 2(x + 3) 57. y = 3x2 58. y = x2 + 2 59. y = −x2

Graph each piecewise function. (LESSON 2.6)

60. f(x) = � 61. g(x) = �Let A = � �, B = � �, and C = � �. Find each

product matrix, if it exists. (LESSON 4.2)

62. AB 63. BA 64. AC 65. CA 66. BC 67. CB

Find a quadratic function to fit each set of points exactly. (LESSON 5.7)

68. (1, −1), (2, −5), (3, 13) 69. (0, 4), (1, 5), (3, 25)

70. Use guess-and-check to find x such that 10x = 50.

−2

4

−6

−1

2

3

2

−2

1

0

7

−5

−1

6

0

4

−8

8

3

−2

10

if 0 ≤ x < 2if 2 ≤ x < 5if 5 ≤ x < 10

x−3x + 8−5

if 0 ≤ x < 5if 5 ≤ x < 10

92x − 1

Look Back

Look Beyond

Ch06 p354-369 1/13/04 3:03 PM Page 369

370 CHAPTER 6

Objectives

● Write equivalent formsfor exponential andlogarithmic equations.

● Use the definitions of exponential andlogarithmic functionsto solve equations.

Logarithmic Functions

Logarithms are used to find unknown exponents in exponential models.

Logarithmic functions define many measurement scales in the sciences,including the pH, decibel, and Richter scales.

Approximating Exponents


Use the table below to complete this Activity.

1. How are the x-values in the table related to the y-values?

2. Use the table above to find the value of x in each equation below.

a. 10x = 1000 b. 10x = �1100�

c. 10x = �10100� d. 10x = 1

3. Make a table of values for y = 10x. Usethe table to approximate the solutionto 10x = 7 to the nearest hundredth.

4. Use a table of values to approximate the solution to 10x = 85 to the nearesthundredth.

Logarithmic functions arewidely used in measurementscales such as the pH scale,which ranges from 0 to 14.

WhyWhy

The pH of an acidic solution is lessthan 7, the pH of a basic solutionis greater than 7, and the pH of aneutral solution is 7.

CHECKPOINT ✔

CHECKPOINT ✔

PROBLEM SOLVING

Substance pH

gastric fluid 1.8

lemon juice 2.2–2.4

vinegar 2.4–3.4

banana 4.8

saliva 6.5–7.5

water 7

egg white 7.6–8.0

Rolaids, Tums 9.9

milk of magnesia 10.5

x −3 −2 −1 0 1 2 3

y = 10x �10100� �1

100� �1

10� 1 10 100 1000

TECHNOLOGYGRAPHICS

CALCULATOR


371LESSON 6.3 LOGARITHMIC FUNCTIONSLESSON 6.3 LOGARITHMIC FUNCTIONS

A table of values for y = 10x can be used to solve equations such as10x = 1000 and 10x = �1

100�

. However, to solve equations such as 10x = 85 or

10x = 2.3, a logarithm is needed. With logarithms, you can write an exponentialequation in an equivalent logarithmic form.

a. Write 53 = 125 in logarithmic form.b. Write log3 81 = 4 in exponential form.

SOLUTION

a. 53 = 125 → 3 = log5 125 3 is the exponent and 5 is the base.b. log3 81 = 4 → 34 = 81 3 is the base and 4 is the exponent.

Copy and complete each column in the table below.

You can evaluate logarithms with a base of 10 by using the key on acalculator.

Solve for 10x = 85 for x. Round your answer to the nearest thousandth.

SOLUTION

Write 10x = 85 in logarithmic form, and use the key.

10x = 85x = log10 85x ≈ 1.9294 Use a calculator.

Solve 10x = �1109�

for x. Round your answer to the nearest thousandth.

LOG

LOG

Equivalent Exponential and Logarithmic Forms

For any positive base b, where b ≠ 1:bx = y if and only if x = log b y

E X A M P L E 1

E X A M P L E 2

Logarithmic form

exponent

base

Exponential form

103 3 == log10

10001000

TRY THIS

TRY THIS

Exponentialform

25 = 32 ? 3−2 = ?

Logarithmicform

? log10 1000 = 3 ? log16 4 = �12�

1�9


CALCULATORBecause 101 = 10 and 10 2 = 100, x ≈ 1.9294 isa reasonable answer.

372 CHAPTER 6

Definition of Logarithmic Function

The inverse of the exponential function y = 10x is x = 10 y. To rewrite x = 10 y

in terms of y, use the equivalent logarithmic form, y = log10 x.

Examine the tables and graphs below to see the inverse relationship between y = 10x and y = log10 x.

The table below summarizes the relationship between the domain and rangeof y = 10x and of y = log10 x.

Describe the graph that results if b = 1 in y = log b x. Is y = log1 x a function?

Because y = logb x is the inverse of the exponential function y = bx and y = logb xis a function, the exponential function y = bx is a one-to-one function. Thismeans that for each element in the domain of an exponential function, there is exactly one corresponding element in the range. For example, if 3x = 32, thenx = 2. This is called the One-to-One Property of Exponents.

One-to-One Property of Exponents

If bx = by, then x = y.

Logarithmic Functions

The logarithmic function y = logb x with base b, or x = by, is the inverseof the exponential function y = bx, where b ≠ 1 and b > 0.

y = log10

x

y = x

y = 10x

2 4 6 108

2

6

4

8

10

(10, 1)

(1, 10)

y

x

x y = log10 x

�10100� −3

�1100� −2

�110� −1

1 0

10 1

100 2

1000 3

x y = 10x

−3 �10100�

−2 �1100�

−1 �110�

0 1

1 10

2 100

3 1000

Function Domain Range

y = 10x all real numbers all positive real numbers

y = log10 x all positive real numbers all real numbers

CRITICAL THINKING

373LESSON 6.3 LOGARITHMIC FUNCTIONS

Find the value of v in each equation.a. v = log125 5 b. 5 = logv 32 c. 4 = log3 v

SOLUTION

Write the equivalent exponential form, and solve for v.a. v = log125 5 b. 5 = logv 32 c. 4 = log3 v

125v = 5 v5 = 32 34 = v(53)v = 5 v5 = 25 81 = v

53v = 51 v = 23v = 1

v = �13�

Find the value of v in each equation.a. v = log4 64 b. 2 = logv 25 c. 6 = log3 v

Recall from Lesson 2.7 that the graph ofy = −f(x) is the graph of y = f(x) reflectedacross the x-axis. The graph of y = log10 xand of its reflection across the x-axis,y = −log10 x, are shown at right.

The function y = −log10 x is used in chemistry to measure pH levels. The pH of a solution describes its acidity. Substances that are more acidic have a lowerpH, while substances that are less acidic, or basic, have a higher pH. The pH of a substance is defined as pH = −log10[H+], where [H+] is the hydrogen ionconcentration of a solution in moles per liter.

The pH of a carbonated soda is 3.

What is [H+] for this soda?

SOLUTION

pH = −log10[H+]3 = −log10[H+] Substitute 3 for pH.

−3 = log10[H+]10−3 = [H+] Write the equivalent exponential

equation.

CHECK

Graph y = −log10 x and y = 3 on the same screen,and find the point of intersection. The windowat right shows x-values between 0 and 0.01.

Thus, there is �10100�, or 0.001, moles of hydrogen

ions in a liter of carbonated soda that has a pH of 3.

Find [H+] for orange juice that has a pH of 3.75.TRY THIS

−2

2 4 6 8

2 y = log10x

y = −log10x

y

x

E X A M P L E 4

E X A M P L E 3

TRY THIS

C O N N E C T I O N

TRANSFORMATIONS


CHEMISTRY

TECHNOLOGYGRAPHICS

CALCULATOR


Apply the One-to-One Property.

1. Describe the relationship between logarithmic functions and exponentialfunctions.

2. State the domain and range of logarithmic functions. How are they relatedto the domain and range of exponential functions?

3. Explain how to approximate the value of x in 2x = 58 by using the tablefeature of a graphics calculator.

4. Write 42 = 16 in logarithmic form. (EXAMPLE 1)

5. Write log5 25 = 2 in exponential form. (EXAMPLE 1)

Solve each equation for x. Round your answers to the nearest

thousandth. (EXAMPLE 2)

6. 10x = 568 7. 10x = �5100�

Find the value of v in each equation. (EXAMPLE 3)

8. v = log7 49 9. 2 = logv 144 10. 2 = log4 v

11. CHEMISTRY The pH of black coffee is 5. What is [H+] for this coffee?(EXAMPLE 4)

Write each equation in logarithmic form.

12. 112 = 121 13. 54 = 625 14. 35 = 243 15. 63 = 216

16. 6−2 = �316�

17. 7−2 = �419�

18. 27�13� = 3 19. 16

�14� = 2

20. (�14�)

−3= 64 21. (�

19�)

−2= 81 22. (�

13�)

2= �

19� 23. (�

12�)

3= �

18�

Write each equation in exponential form.

24. log6 36 = 2 25. log10 1000 = 3 26. log10 0.001 = −3

27. log10 0.1 = −1 28. 3 = log9 729 29. 3 = log7 343

30. log3 �811�

= −4 31. log2 �312�

= −5 32. −2 = log2 �14�

33. −3 = log3 �217�

34. log121 11 = �12� 35. log144 12 = �

12�

Find the approximate value of each logarithmic expression.

36. log10 1026 37. log10 79 38. log10 8 39. log10 21,050

40. log10 0.08 41. log10 0.9 42. log10 0.002 43. log10 0.00013

374 CHAPTER 6

ExercisesExercises

Communicate



Practice and Apply

Solve each equation for x. Round your answers to the nearest

hundredth.

44. 10x = 31 45. 10x = 12 46. 10x = 7210

47. 10x = 3588 48. 10x = 1.498 49. 10x = 1.89

50. 10x = 0.0054 51. 10x = 0.035 52. 10x = �439�

53. 10x = �10185� 54. 10x = �7.�4� 55. 10x = �

�510�0��

Find the value of v in each equation.

56. v = log10 1000 57. v = log4 64 58. v = log7 343

59. v = log17 289 60. v = log3 3 61. v = log7 7

62. v = log10 0.001 63. v = log10 0.01 64. v = log2 �14�

65. v = log10 �1100�

66. v = log4 1 67. v = log9 1

68. 3 = log6 v 69. 2 = log7 v 70. 1 = log5 v

71. 1 = log3 v 72. �12� = log9 v 73. �

13� = log8 v

74. −2 = log6 v 75. −3 = log4 v 76. 0 = log13 v

77. 0 = log2 v 78. logv 16 = 2 79. logv 125 = 3

80. logv 9 = �12� 81. logv 4 = �

13� 82. logv �1

16�

= −4

83. logv �18� = −3 84. logv 216 = 3 85. logv 243 = 5

86. Graph f(x) = 3x along with f −1. Make a table of values that illustrates therelationship between f and f −1.

87. Graph f(x) = 3−x along with f −1. Make a table of values that illustrates therelationship between f and f −1.

Find the value of each expression.

88. log27 �3� 89. log2 16�2� 90. log �12� 8

TRANSFORMATIONS Let f(x) = log10x. For each function, identify the


91. g(x) = 3 log10 x 92. g(x) = −5 log10 x

93. g(x) = �12� log10 x + 1 94. g(x) = 0.25 log10 x − 2

95. g(x) = −log10(x − 2) 96. g(x) = log10(x + 5) − 3

CHEMISTRY Calculate [H+] for each of the following:

97. household ammonia with a pH of about 10

98. distilled water with a pH of 7

99. human blood with a pH of about 7.4

100. CHEMISTRY How much greater is [H+] for lemon juice, which has a pH of2.1, than [H+] for water, which has a pH of 7.0?

C H A L L E N G E S

pH paper turns red in anacidic solution, 0 < pH < 7;the paper turns green in aneutral solution, indicatinga pH of 7; and the paperturns blue in a basicsolution, 7 < pH < 14.

375LESSON 6.3 LOGARITHMIC FUNCTIONS



Stomachacid,

lemons

Batteryacid

Tomatoes,bananas

Blackcoffee

Purewater

Bakingsoda

Handsoap

Householdammonia

Lye(drain

cleaner)

10 2 3 4 5 6 7 8 9 10 11 12 13 14

Look Back

Look Beyond

101. PHYSICS Earth’s atmosphere islike an “ocean” of air with theupper layers of air pressing down on the lower layers of air. The weight of the layers of air creates atmospheric air pressure. At sea level (altitudeof zero), the average air pressure is about 14.7 pounds per square inch.The air pressure, P, decreases with altitude, a, in feet according to thefunction P = 14.7(10)−0.000018a. Find the altitude that corresponds to theair pressure commonly found in commercial airplanes, 11.82 poundsper square inch.

102. Write a linear equation for a line with a slope of 4 and a y-intercept of 3.(LESSON 1.2)

State the property that is illustrated in each statement. All variables

represent real numbers. (LESSON 2.1)

103. 1 • (5xy) = 5xy 104. (2 + z) + y = 2 + (z + y)

105. 2(3x) = 3x(2) 106. −x + 0 = −x

107. �2a

� • �2a� = 1, where a ≠ 0 108. −3 + x = x + (−3)

109 Find the inverse of the matrix � �. (LESSON 4.3)

110. Solve the quadratic equation x2 − 6x + 9 = 0. (LESSONS 5.2 AND 5.4)

111. State the two solutions of the equation x2 + 1 = 0. (LESSON 5.6)

112. If an interest rate is 7.3%, what is the multiplier? (LESSON 6.1)

113. Calculate log2 2 + log2 8 and log2 32 − log2 2. Then compare these valueswith the value of log2 16.

3−4

21

376 CHAPTER 6


377LESSON 6.4 PROPERTIES OF LOGARITHMIC FUNCTIONS

Objectives

● Simplify and evaluateexpressions involvinglogarithms.

● Solve equationsinvolving logarithms.

In the seventeenth century, a Scottish mathematician named John Napierdeveloped methods for efficiently performing calculations with large numbers.He found a method for finding the product of two numbers by adding twocorresponding numbers, which he called logarithms.

John Napier’s contributions to mathematics are contained in two essays:Mirifici Logarithmorum Canonis Descriptio (Description of the MarvelousCanon of Logarithms), published in 1614, and Mirifici LogarithmorumCanonis Constructio (Construction of the Marvelous Canon of Logarithms),published in 1619, two years after his death.

Properties ofLogarithmic Functions

The properties of logarithmsallow you to simplify logarithmicexpressions, which makesevaluating the expressions easier.

WhyWhy

John Napier(1550–1617)

Title page and calculations fromNapier’s Mirifici LogarithmorumCanonis Descriptio

378 CHAPTER 6

The Product, Quotient, and Power Properties of Exponents are as follows:

am • an = am + n Product Property

�aa

m

n� = am − n Quotient Property

(am)n = am • n Power Property

Each property of exponents has a corresponding property of logarithms.

Exploring Properties of Logarithms

You will need: no special tools

Use the following table to complete the activity:

1. The expression log2 (2 • 4) can be written as log2 8. Use this fact and thetable above to evaluate each expression below.

a. log2(2 • 4) = n?n and log2 2 + log2 4 = n?n

b. log2(2 • 8) = n?n and log2 2 + log2 8 = n?n

c. log2(2 • 16) = n?n and log2 2 + log2 16 = n?n

d. log2(2 • 32) = n?n and log2 2 + log2 32 = n?n

2. In Step 1, how is the first expression in each pair related to the secondexpression? Use this pattern to make a conjecture about log2 (a • b).

3. The expression log2 �126� can be written as log2 8. Use this fact and the

table above to evaluate each expression below.

a. log2 �126� = n?n and log2 16 − log2 2 = n?n

b. log2 �63

42�

= n?n and log2 64 − log2 32 = n?n

c. log2 �382� = n?n and log2 32 − log2 8 = n?n

d. log2 �84� = n?n and log2 8 − log2 4 = n?n

4. In Step 3, how is the first expression in each pair related to the secondexpression? Use this pattern to make a conjecture about log2 �

ab�.

The patterns explored in the Activity illustrate the Product and QuotientProperties of Logarithms given below.

Product and Quotient Properties of Logarithms

For m > 0, n > 0, b > 0, and b ≠ 1:Product Property logb(mn) = logb m + logb nQuotient Property logb �

mn� = logb m − logb n

CHECKPOINT ✔

CHECKPOINT ✔

Product and Quotient Properties of Logarithms

x 2 4 8 16 32 64 128

y = log 2 x 1 2 3 4 5 6 7

E X A M P L E 1


You can use the Product and Quotient Properties of Logarithms to evaluatelogarithmic expressions. This is shown in Example 1.

Given log2 3 ≈ 1.5850, approximate the value of each expression below byusing the Product and Quotient Properties of Logarithms.

a. log2 12 b. log2 1.5

SOLUTION

a. log2 12 = log2 (2 • 2 • 3) b. log2 1.5 = log2 �32�

= log2 2 + log2 2 + log2 3 = log2 3 − log2 2≈ 1 + 1 + 1.5850 ≈1.5850 − 1≈ 3.5850 ≈ 0.5850

Given that log2 3 = 1.5850, approximate each expression below by using theProduct and Quotient Properties of Logarithms.

a. log2 18 b. log2 �34�

Example 2 demonstrates how to use the properties of logarithms to rewrite alogarithmic expression as a single logarithm.

Write each expression as a single logarithm. Then simplify, if possible.a. log3 10 − log3 5 b. logb u + logb v − logb uw

SOLUTION

a. log3 10 − log3 5 = log3 �150� b. logb u + logb v − logb uw = logb uv − logb uw

= log3 2 = logb �uuwv�

= logb �wv

�

Write each expression as a single logarithm. Then simplify if possible.a. log4 18 − log4 6 b. logb 4x − logb 3y + logb y

The Power Property of Logarithms

Examine the process of rewriting the expression logb(a4).

logb(a4) = logb(a • a • a • a)= logb a + logb a + logb a + logb a= 4 • logb a

This illustrates the Power Property of Logarithms given below.

Power Property of Logarithms

For m > 0, b > 0, b ≠ 1, and any real number p:logb mp = p logb m

E X A M P L E 2

TRY THIS

TRY THIS

380 CHAPTER 6

In Example 3, the Power Property of Logarithms is used to simplify powers.

Evaluate log5 254.

SOLUTION

log5 254 = 4 log5 25 Use the Power Property of Logarithms.= 4 • 2= 8

Evaluate log3 27100.

Recall from Lesson 2.5 that functions f and g are inverse functions if and onlyif ( f ° g)(x) = x and (g ° f )(x) = x. The functions f(x) = logb x and g(x) = bx areinverses, so ( f ° g)(x) = logb bx = x and (g ° f )(x) = blogb x = x.

Evaluate each expression.a. 3log3 4 + log5 25 b. log2 32 − 5log5 3

SOLUTION

a. 3log3 4 + log5 25 b. log2 32 − 5log5 3

= 4 + log5 52 = log2 25 − 3= 4 + 2 = 5 − 3= 6 = 2

Evaluate each expression.a. 7log7 11 − log3 81 b. log8 85 + 3log3 8

Verify the Exponential-Logarithmic Inverse Properties by using only theequivalent exponential and logarithmic forms given on page 371.

Because exponential functions and logarithmic functions are one-to-onefunctions, for each element in the domain of y = log x, there is exactly onecorresponding element in the range of y = log x.

One-to-One Property of Logarithms

If logb x = logb y, then x = y.

Exponential-Logarithmic Inverse Properties

For b > 0 and b ≠ 1:logb bx = x and b logb x = x for x > 0

TRY THIS

E X A M P L E 3

E X A M P L E 4

TRY THIS

CRITICAL THINKING

Exponential-Logarithmic Inverse Properties


Solve log3(x2 + 7x − 5) = log3(6x + 1) for x. Check your answers.

SOLUTION

log3(x2 + 7x − 5) = log3(6x + 1)x2 + 7x − 5 = 6x + 1 Use the One-to-One Property of Logarithms.

x2 + x − 6 = 0(x − 2)(x + 3) = 0

x = 2 or x = −3 Use the Zero Product Property.

CHECK

Let x = 2. Let x = −3.log3(x2 + 7x − 5) ?

= log3(6x + 1) log3(x2 + 7x − 5) ?= log3(6x + 1)

log3 13 = log3 13 log3(−17) = log3(−17)True Undefined

Since the domain of a logarithmic function excludes negative numbers, thesolution cannot be −3. Therefore, the solution is 2.

1. Given that log10 5 ≈ 0.6990, explain how to approximate the values oflog10 0.005 and log10 500.

2. Explain how to write an expression such as log7 32 − log7 4 as a singlelogarithm.

3. Explain how to evaluate 4log4 8 and log2 27. Include the names of theproperties you would use.

4. Explain why you must check your answers when solving an equation suchas log2 3x = log2(x + 4) for x.

Given log3 7 ≈ 1.7712, approximate the value for each logarithm by

using the Product and Quotient Properties of Logarithms. (EXAMPLE 1)

5. log3 49 6. log3 �37�

Write each expression as a single logarithm. Then simplify, if possible.

(EXAMPLE 2)

7. log3 x − log3 y + log3 z 8. log2 3 + log2 6 − log2 10

Evaluate each expression. (EXAMPLES 3 AND 4)

9. log4 168 10. 3log3 12 11. log7 73

12. Solve log3 x = log3(2x − 4) for x, and check your answers. (EXAMPLE 5)

E X A M P L E 5

ExercisesExercises

Communicate


382 CHAPTER 6

Write each expression as a sum or difference of logarithms. Then

simplify, if possible.

13. log8(5 • 8) 14. log2 8xy 15. log3 �9x

� 16. log4 �3x2�

Use the values given below to approximate the value of each

logarithmic expression in Exercises 17–28.

17. log4 15 18. log2 35 19. log2 28

20. log4 12 21. log4 60 22. log2 105

23. log10 830 24. log10 0.0083 25. log4 �35�

26. log2 �170�

27. log4 �54� 28. log2 �

27�

Write each expression as a single logarithm. Then simplify, if possible.

29. log2 5 + log2 7 30. log4 8 + log4 2

31. log3 45 − log3 9 32. log2 14 − log2 7

33. log2 5 + log2 x − log2 10 34. log3 x + log3 4 − log3 2

35. log7 3x − log7 9x + log7 6y 36. log5 6s − log5 s + log5 4t

37. 5 log2 m − 2 log2 n 38. 7 log3 y − 4 log3 x

39. 4 logb m + �12� logb n − 3 logb 2p 40. �

12� logb 3c + �

12� logb 4d − 2 logb 5e

41. 1 − 2 log7 x 42. 2 + 4 log3 x

Evaluate each expression.

43. 3log3 8 44. 9log9 2 45. log4 45

46. log10 102 47. 7log7 9 + log2 8 48. 5log5 7 + log3 9

49. log9 911 − log4 64 50. log3 35 + log5 125 51. 6log6 3 − log5 �215�

52. 2log2 3 + log6 �316�

53. log3 �19� − 2log2 3 54. log2 �

18� − 4log4 7

Solve for x, and check your answers. Justify each step in the solution process.

55. log2 7x = log2(x2 + 12) 56. log5(3x2 − 1) = log5 2x

57. logb(x2 − 15) = logb(6x + 1) 58. log10(5x − 3) − log10(x2 + 1) = 0

59. 2 loga x + loga 2 = loga(5x + 3) 60. logb(x2 − 2) + 2 logb 6 = logb 6x

61. 2 log3 x + log3 5 = log3(14x + 3) 62. log5 2 + 2 log5 t = log5(3 − t)

State whether each equation is always true, sometimes true, or never

true. Assume that x is a positive real number.

63. log3 9 = 2 log3 3 64. log2 8 − log2 2 = 2 65. log x2 = 2 log x

66. log x − log 5 = log �5x

� 67. �l

l

o

o

g

g

3

x� = log 3 − log x 68. log(x − 2) = �

l

l

o

o

g

g

2

x�

69. �12� log x = log �x� 70. log 12x = 12 log x 71. log3 x + log3 x = log3 2x

Practice and Apply

log2 7 ≈ 2.8074 log2 5 ≈ 2.3219 log4 5 ≈ 1.1610

log4 3 ≈ 0.7925 log2 3 ≈ 1.5850 log10 8.3 ≈ 0.9191


for Exercises 13–16,29–42

Ch06 p370-384 4/23/02 8:30 PM Page 382


Solve each equation.

72. log4(log3 x) = 0 73. log6[log5(log3 x)] = 0

74. HEALTH The surface area of a personis commonly used to calculatedosages of medicines. The surfacearea of a child is often calculated withthe following formula, where S is thesurface area in square centimeters, Wis the child’s weight in kilograms, andH is the child’s height in centimeters.

log10 S = 0.425 log10 W + 0.725 log10 H + log10 71.84

Use the properties of logarithms to write a formula for S withoutlogarithms.

75. PHYSICS Atmospheric air pressure, P, in pounds per square inch and altitude, a, in feet are related by the logarithmic equationa = −55,555.56 log10 �14

P.7�. Use properties of logarithms to find how much

greater the air pressure at the top of Mount Whitney in the United Statesis compared with the air pressure at thetop of Mount Everest on the border ofTibet and Nepal. The altitude of MountEverest is 29,028 feet, and the altitude ofMount Whitney is 14,495 feet. (Hint:Find the ratio of the air pressures.)

Minimize each objective function under the given constraints.

(LESSON 3.5)

76. Objective function: C = 2x + 5y 77. Objective function: C = x + 4y

Constraints: � Constraints: �Write the matrix equation that represents each system. (LESSON 4.4)

78. � 79. � 80. �x + y + z = 18

�14�x + �

12�y + �

13�z = 6

2y + 3z = 33

3x + 2y − z = 75x + 3y − 2z = −123y − z + 2x = −5

2x − y = 53x + 4y = −3

x − y ≥ 127y − x ≤ 12x ≥ 0y ≥ 0

x + 5y ≥ 8y − 3x ≤ 14x ≥ 0y ≥ 0


C H A L L E N G E

Mount Whitney14, 495 feet

Sea level

Mount Everest29,028 feet

Look Back

Look Beyond

A C T I V

I TY

PO

RTFOLIO

In this activity, you will use the CBL to collectdata as a warm probe cools in air.

1. First record the air temperature forreference. Then place the temperature probein hot water until it reaches a reading of atleast 60˚C. Remove the probe from thewater and record the readings as in Step 1 ofthe Portfolio Activity on page 361.

2. a. Using the regression feature on your calculator, find linear, quadratic, and exponential functions that model thisdata. (Use the variable t for time inseconds.)

b. Use each function to predict the temperature of the probe after 2 minutes (120 seconds). Compare the predictionswith the actual 2-minute reading.

c. Discuss the usefulness of each functionfor modeling the cooling process.

3. Create a new function to model the coolingprocess by performing the steps below.a. Subtract the air temperature from each

temperature recorded in your data list.Store the resulting data values in a new list.

b. Use the exponential regression feature onyour calculator to find an exponentialfunction of the form y = a • bt thatmodels this new data set.

c. Add the air temperature to the functionyou found in part b. Graph the resultingfunction, y = a • bt + c, which will becalled the approximating function.

d. Repeat parts b and c from Step 2 with theapproximating function.

Save your data and results to use in the lastPortfolio Activity.


384 CHAPTER 6

Write the augmented matrix for each system of equations. (LESSON 4.5)

81. � 82. � 83. �84. INVESTMENTS An investment of $100 earns an annual interest rate of 5%.

Find the amount after 10 years if the interest is compounded annually,quarterly, and daily. (LESSON 6.2)

85. INVESTMENTS Find the final amount after 8 years of a $500 investmentthat is compounded semiannually at 6%, 7%, and 8% annual interest.(LESSON 6.2)

86 e is an irrational number between 2 and 3. The expression log e x iscommonly written as ln x. Use the key to solve 2e3x = 5 for x to thenearest hundredth.

LN

0.5x + 0.3y = 2.2−8.5y + 1.2z = −24.43.3z + 1.3x = 29

3x − 6y + 3z = 4x − 2y = 1 − z2x − 4y + 2z = 5

−3x + 2y = 114x = 5 − y


388 CHAPTER 6

The Activity below leads to a method for evaluating logarithmic expressionswith bases other than 10.

Exploring Change of Base

You will need: a scientific calculator

1. Write 3x = 81 as a logarithmic expression for x in base 3.

2. Write 3x = 81 as a logarithmic expression for x in base 10.(Hint: Refer to Example 3.)

3. Set your expressions for x from Steps 1 and 2 equal to each other.

4. Write bx = y in logarithmic form. Then solve bx = y for x, and give theresult as a quotient of logarithms. Set your two resulting expressionsequal to each other.

The answer to Step 4 in the Activity suggests a change-of-base formula, shownbelow, for writing equivalent logarithmic expressions with different bases.

Write log9 27 as a base 3 expression.

You can use the change-of-base formula to change a logarithmic expression ofany base to base 10 so that you can use the key on a calculator. This isshown in Example 4.

Evaluate log7 56. Round your answer to the nearest hundredth.

SOLUTION

Use the change-of-base formula to change from base 7 to base 10.

log7 56 = �lloogg

576

�

≈ �10

.

.78

44

85�

≈ 2.07 Use a calculator to evaluate.

Evaluate log8 36. Round your answer to the nearest hundredth.

Use the change-of-base formula to justify each formula below.

a. (loga b)(logb c) = loga c b. loga b = �log1

b a�

LOG

Change-of-Base Formula

For any positive real numbers a ≠ 1, b ≠ 1, and x > 0:

logb x = �lloo

gg

a

a

bx

�

CHECKPOINT ✔

CHECKPOINT ✔

E X A M P L E 4

TRY THIS

CRITICAL THINKING


CALCULATOR

389LESSON 6.5 APPLICATIONS OF COMMON LOGARITHMS

1. Explain why a common logarithmic function is appropriate to use for thedecibel scale of sound intensities.

2. Describe the steps you would take to solve 6x = 39 for x.

3. Explain how to evaluate log4 29 by using a calculator.

4. PHYSICS Suppose that a soft whisper is about 75 times as loud as thethreshold of hearing, I0. Find the relative intensity, R, of this whisper indecibels. (EXAMPLE 1)

5. PHYSICS The relativeintensity, R, of a loud siren isabout 130 decibels. Comparethe intensity of this sirenwith the threshold of hearing,I0. (EXAMPLE 2)

Solve each exponential

equation for x. Round your

answers to the nearest

hundredth. (EXAMPLE 3)

6. 8x = 47. 4x = 72

Evaluate each logarithmic

expression. Round your

answers to the nearest

hundredth. (EXAMPLE 4)

8. log2 469. log5 2

Solve each equation. Round your answers to the nearest hundredth.

10. 4x = 17 11. 2x = 49 12. 7x = 908

13. 8x = 240 14. 3.5x = 28 15. 7.6x = 64

16. 25x = 0.04 17. 3x = 0.26 18. 2−x = 0.045

19. 7−x = 0.022 20. 3x = 0.45 21. 5x = 1.29

22. 2x +1 = 30 23. 3x − 6 = 81 24. 11 − 6x = 3

25. 67 − 2x = 39 26. 8 + 3x = 10 27. 1 + 5x = 360

ExercisesExercises

Communicate


Practice and Apply




390 CHAPTER 6

Evaluate each logarithmic expression to the nearest hundredth.

28. log4 92 29. log6 87 30. log6 18

31. log3 15 32. log6 3 33. log5 2

34. log9 4 35. log8 3 36. log4 0.37

37. log9 1.43 38. log �13� 9 39. log �12� 8

40. log8 �14� 41. log7 �5

10�

42. 8 − log2 64

43. 1 − log5 21 44. 9 + log3 27 45. 4 + log5 125

46. Prove that log(bn) x = �n1

� logb x is true.

47. PHYSICS The sound of a leaf blower is about 1010.5 times the intensity ofthe threshold of hearing, I0. Find the relative intensity, R, of this leafblower in decibels.

48. PHYSICS The sound of a conversation is about 350,000 times the intensityof the threshold of hearing, I0. Find the relative intensity, R, of thisconversation in decibels.

49. PHYSICS Suppose that the relative intensity, R, of a rock band is about 115 decibels. Compare the intensity of this band with that of the thresholdof hearing, I0.

50. PHYSICS The relative intensity, R, of an automobile engine is about 55 decibels. Compare the intensity of this engine with that of thethreshold of hearing, I0.

51. PHYSICS Suppose that background music is adjusted to an intensity that is 1000 times as loud as the threshold of hearing. What is the relativeintensity of the music in decibels?

52. PHYSICS Suppose that a burglar alarm has a rating of 120 decibels.Compare the intensity of this decibel rating with that of the threshold ofhearing, I0.

53. PHYSICS Simon Robinson set the world record for the loudest scream byproducing a scream of 128 decibels at a distance of 8 feet and 2 inches.Compare the intensity of this decibel rating with that of the threshold ofhearing, I0. [Source: The Guinness Book of World Records, 1997]

54. PHYSICS A small jet engine produces a sound whose intensity is one billiontimes as loud as the threshold of hearing. What is the relative intensity ofthe engine’s sound in decibels?

C H A L L E N G E


Activities OnlineGo To: go.hrw.comKeyword:MB1 Stock Closings

392 CHAPTER 6

Objectives

● Evaluate naturalexponential andnatural logarithmicfunctions.

● Model exponentialgrowth and decayprocesses.

The Natural Base, e

The natural base, e, is used to estimate the ages of artifacts and to calculateinterest that is compounded continuously. Recall from Lesson 6.2 the

compound interest formula, A(t) = P(1 + �nr

�)nt

, where P is the principal, r is

the annual interest rate, n is the number of compounding periods per year,and t is the time in years. This formula is used in the Activity below.

Investigating the Growth of $1

You will need: a scientific calculator

1. Copy and complete the table below to investigate the growth of a $1 investment that earns 100% annual interest (r = 1) over 1 year (t = 1)as the number of compounding periods per year, n, increases. Use acalculator, and record the value of A to five places after the decimal point.

2. Describe the behavior of the sequence of numbers in the Value column.

As n becomes very large, the value of 1(1 + �n1

�)n

approaches the number

2.71828 . . . , named e. Because e is an irrational number like π, its decimalexpansion continues forever without repeating patterns.

The exponential function withbase e and its inverse, the naturallogarithmic function, have a widevariety of real-world applications.For example, these functions areused to estimate the ages of artifactsfound at archaeological digs.

WhyWhy


INVESTMENTS

CHECKPOINT ✔

Compounding schedule n 1(1 + �n1

�)n

Value, A

annually 1 1(1 + �11�)1

2.00000

semiannually 2 1(1 + �12�)2

quarterly 4

monthly 12

daily 365

hourly

every minute

every second

The model belowshows the embryoinside an 18-inchdinosaur egg, thelargest known.

393

The Natural Exponential Function

The exponential function with base e, f(x) = e x,is called the natural exponential function and eis called the natural base. The function f(x) = e x

is graphed at right. Notice that the domain is all real numbers and the range is all positive real numbers.

What is the y-intercept of the graph of f(x) = e x ?

Natural exponential functions model a variety ofsituations in which a quantity grows or decayscontinuously. Examples that you will solve in thislesson include continuous compounding interestand continuous radioactive decay.

Evaluate f(x) = e x to the nearest thousandth for each value of x below.

a. x = 2 b. x = �12� c. x = −1

SOLUTION

a. f(2) = e2 b. f (�12�) = e

�12�

c. f(−1) = e −1

≈ 7.389 ≈ 1.649 ≈ 0.368

CHECK

Use a table of values for y = e x or a graph of y = e x to verify your answers.

Evaluate f(x) = e x to the nearest thousandth for x = 6 and x = − �13�.

Many banks compound the interest on accounts daily or monthly. However,some banks compound interest continuously, or at every instant, by using the continuous compounding formula, which includes the number e.

Continuous Compounding Formula

If P dollars are invested at an interest rate, r, that is compoundedcontinuously, then the amount, A, of the investment at time t is given by

A = Pert.

f(x) = ex

(1, e)

1

20

40

60

80

y

x2 3 4

E X A M P L E 1

TRY THIS

LESSON 6.6 THE NATURAL BASE, e

CHECKPOINT ✔

TECHNOLOGYGRAPHICS

CALCULATOR


(−1, ≈0.368)(0.5, ≈1.649)

(2, ≈7.389)

An investment of $1000 earns an annual interest rate of 7.6%.

Compare the final amounts after 8 years for interest compounded quarterlyand for interest compounded continuously.

SOLUTION

Substitute 1000 for P, 0.076 for r, and 8 for t in the appropriate formulas.

Interest that is compounded continuously results in a final amount that isabout $10 more than that for the interest that is compounded quarterly.

Find the value of $500 after 4 years invested at an annual interest rate of 9%compounded continuously.

The Natural Logarithmic Function

The natural logarithmic function,y = loge x, abbreviated y = ln x, is theinverse of the natural exponentialfunction, y = e x. The function y = ln x is graphed along with y = e x at right.

State the domain and range of y = e x

and of y = ln x.

Evaluate f(x) = ln x to the nearest thousandth for each value of x below.

a. x = 2 b. x = �12� c. x = −1

SOLUTION

a. b. c.

CHECK

Use a table of values for y = ln x or a graph of y = ln x to verify your answers.

f(−1) = ln(−1) is undefined.

f (�12�) = ln �

12�

≈ −0.693f(2) = ln 2

≈ 0.693

394 CHAPTER 6

E X A M P L E 2


INVESTMENTS

Compounded quarterly Compounded continuously

A = P(1 + �nr

�)ntA = Pert

A = 1000(1 + �0.0

476�)

4 • 8A = 1000e0.076 • 8

A ≈ 1826.31 A ≈ 1836.75

TRY THIS

2 4 6 8

2

4

6

8

y = x

y = e x

y = ln x

1

3

5

7

1 3 5 7

y

xCHECKPOINT ✔

E X A M P L E 3

TECHNOLOGYGRAPHICS

CALCULATOR


(2, ≈0.693)

Nonpositive numbers are notin the domain of y = ln x.

(2, ≈−0.693)

395LESSON 6.6 THE NATURAL BASE, e

The natural logarithmic function can be used to solve an equation of the formA = Pert for the exponent t in order to find the time it takes for an investmentthat is compounded continuously to reach a specific amount. This is shown inExample 4.

How long does it take for an investment to double at an annual interest rateof 8.5% compounded continuously?

SOLUTION

Use the formula A = Pert with r = 0.085.

A = Pe0.085t

2 • P = Pe0.085t When the investment doubles, A = 2 • P.2 = e0.085t

ln 2 = ln e0.085t Take the natural logarithm of both sides.ln 2 = 0.085t Use the Exponential-Logarithmic Inverse Property.

t = �0l.n08

25�

t ≈ 8.15

CHECK

Graph y = e0.085x and y = 2, and find the point of intersection.

Notice that the graph of y = e0.085x is ahorizontal stretch of the function y = e x

by a factor of �0.0185�, or almost 12.

Thus, it takes about 8 years and 2 months to double an investment at an annual interest rate of 8.5% compounded continuously.

How long does it take for an investment to triple at an annual interest rate of7.2% compounded continuously?

Explain why the time required for the value of an investment to double ortriple does not depend on the amount of principal.

E X A M P L E 4

TRY THIS

CRITICAL THINKING

Radioactive Decay

Most of the carbon found in the Earth’s atmosphere is the isotope carbon-12,but a small amount is the radioactive isotope carbon-14. Plants absorb carbondioxide from the atmosphere, and animals obtain carbon from the plants theyconsume. When a plant or animal dies, the amount of carbon-14 it containsdecays in such a way that exactly half of its initial amount is present after 5730years. The function below models the decay of carbon-14, where N0 is theinitial amount of carbon-14 and N(t) is the amount present t years after theplant or animal dies.

N(t) = N0e−0.00012t

TECHNOLOGYGRAPHICS

CALCULATOR


PROBLEM SOLVING

396 CHAPTER 6

Example 5 shows how radiocarbon dating is used to estimate the age of anarchaeological artifact.

Suppose that archaeologists find scrolls and claim that they are 2000 years old.Tests indicate that the scrolls contain 78% of their original carbon-14.

Could the scrolls be 2000 years old?

SOLUTION

Since the scrolls contain 78% of their original carbon-14, substitute 0.78N0

for N(t).

N(t) = N0e−0.00012t

0.78N0 = N0e−0.00012t Substitute 0.78N0 for N(t).0.78 = e−0.00012t

ln 0.78 = −0.00012t Take the natural logarithm of each side.−0.00012t = ln 0.78

t = �−l0n.0

00.07

18

2�

t ≈ 2070.5

Thus, it appears that the scrolls are about 2000 years old.

1. Compare the natural and exponential logarithmic functions with the base-10 exponential and logarithmic functions.

2. Give a real-world example of an exponential growth function and of anexponential decay function that each have the base e.

3. State the continuous compounding formula, and describe what eachvariable represents.

4. Describe how the continuous compounding formula can representcontinuous growth as well as continuous decay.

Evaluate f(x) = ex to the nearest thousandth for each value of x.

(EXAMPLE 1)

5. x = 3 6. x = 3.5

7. INVESTMENTS An investment of $1500 earns an annual interest rate of8.2%. Compare the final amounts after 5 years for interest compoundedquarterly and for interest compounded continuously. (EXAMPLE 2)

E X A M P L E 5

ExercisesExercises

Communicate




ARCHAEOLOGY




Evaluate f(x) = ln x to the nearest thousandth for each value of x.

(EXAMPLE 3)

8. x = 5 9. x = 2.5

10. INVESTMENTS How long does it take an investment to double at an annualinterest rate of 7.5% compounded continuously? (EXAMPLE 4)

11. ARCHAEOLOGY A piece of charcoal from an ancient campsite is found in anarchaeological dig. It contains 9% of its original amount of carbon-14.Estimate the age of the charcoal. (EXAMPLE 5)

Evaluate each expression to the nearest thousandth. If the expression is

undefined, write undefined.

12. e6 13. e9 14. e1.2 15. e 3.4

16. 2e0.3 17. 3e0.05 18. 2e −0.5 19. 3e −0.257

20. e 21. e�14�

22. ln 3 23. ln 7

24. ln 10,002 25. ln 99,999 26. ln 0.004 27. ln 0.994

28. ln �15� 29. ln �5� 30. ln(−2) 31. ln(−3)

For Exercises 32–35, write the expressions in ascending order.

32. e2, e5, ln 2, ln 5 33. e, e0, ln 1, ln �12�

34. e2.5, ln 2.5, 102.5, log 2.5 35. e1.3, ln 1.3, 101.3, log 1.3

State whether each equation is always true, sometimes true, or never true.

36. e5x • e3 = e15x 37. (e4x)3= e12x 38. e6x −4 = e6x • e−4 39. �

ee

8

4

x� = e2x

Simplify each expression.

40. e ln 2 41. e ln 5 42. e 3ln 2 43. e 2 ln 5

44. ln e3 45. ln e 4 46. 3 ln e2 47. 2 ln e 4

Write an equivalent exponential or logarithmic equation.

48. ex = 30 49. ex = 1 50. ln 2 ≈ 0.69

51. ln 5 ≈ 1.61 52. e�13� ≈ 1.40 53. e0.69 ≈ 1.99

Solve each equation for x by using the natural logarithm function.

Round your answers to the nearest hundredth.

54. 35x = 30 55. 1.3x = 8 56. 3−3x = 17

57. 362x = 20 58. 0.42−x = 7 59. 2− �

13�x = 10

60. Sketch f(x) = e x for −1 ≤ x ≤ 2. A line that intersects a curve at only onepoint is called a tangent line of the curve.a. Sketch lines that are tangent to the graph of f(x) = e x at x = 0.5, x = 0,

x = 1, and x = 2.b. Find the approximate slope of each tangent line. Compare the slope of

each tangent line with the corresponding y-coordinate of the pointwhere the tangent line intersects the graph.

c. Make a conjecture about the slope of f(x) = e x as x increases.

�2�

Practice and Apply


C H A L L E N G E

TRANSFORMATIONS Let f(x) = ex. For each function, describe the


61. g(x) = 6e x + 1 62. g(x) = 0.75e x − 4

63. g(x) = 0.25e (4x + 4) 64. g(x) = 3e (2x − 4)

TRANSFORMATIONS Let f(x) = ln x. For each function, describe the


65. g(x) = 3 ln(x + 1) 66. g(x) = −2 ln(x − 1)

67. g(x) = 0.5 ln(5x) − 2 68. g(x) = 5 ln(0.25x) − 1

69. TRANSFORMATIONS The graphs of f(x) = e −2x, g(x) = e −x, h(x) = e x, and i(x) = e2x are shown on the same coordinate plane at left. Whattransformations relate each function, f, g, and i, to h?

70. TRANSFORMATIONS For f(x) = ex, describe how each transformation affectsthe domain, range, asymptotes, and y-intercept.a. a vertical stretch b. a horizontal stretchc. a vertical translation d. a horizontal translation

71. TRANSFORMATIONS For f(x) = ln x, describe how each transformationaffects the domain, range, asymptotes, and x-intercept.a. a vertical stretch b. a horizontal stretchc. a vertical translation d. a horizontal translation

72. PHYSICS The amount of radioactive strontium-90 remaining after t yearsdecreases according to the function N(t) = N0e −0.0238t. How much of a 40-gram sample will remain after 25 years?

73. ECONOMICS The factory sales of pagers from 1990 through 1995 can bemodeled by the function S = 116e0.18t, where t = 0 in 1990 and S representsthe sales in millions of dollars. [Source: Electronic Market Data Book]a. According to this function, find the factory sales of pagers in 1995 to

the nearest million.b. If the sales of pagers continued to increase at the same rate, when

would the sales be double the 1995 amount?

74. INVESTMENTS Compare the growth of an investment of $2000 in twodifferent accounts. One account earns 3% annual interest, the other earns5% annual interest, and both are compounded continuously over 20 years.

75. ARCHAEOLOGY A wooden chest is found and is said to befrom the second century B.C.E. Tests on a sample of woodfrom the chest reveal that it contains 92% of its originalcarbon-14. Could the chest be from the second centuryB.C.E.?

398 CHAPTER 6

C O N N E C T I O N S

−2

4

6

8

f(x) = e−2x

g(x) = e−x

i(x) = e2x

h(x) = ex

2

2


Sale

s (i

n m

illi

ons

of $

)

1

2

3

1986

0.65

1991

1.5

1996

3.2

Basketball Backboard

Sales


76. BUSINESS Sales of home basketball backboards from 1986 to 1996 can bemodeled by S = 0.65e 0.157t, where S is the sales in millions of dollars, t istime in years, and t = 0 in 1986. [Source: Huffy Sports]a. Use this model to estimate the sales of backboards in 1997 to the

nearest thousand.b. If the sales of basketball backboards continued to increase at the same

rate, when would the sales of basketball backboards be double theamount of 1996?

INVESTMENTS For Exercises 77−79, assume that all interest rates are

compounded continuously.

77. How long will it take an investment of $5000 to double if the annualinterest rate is 6%?

78. How long will it take an investment to double at 10% annual interest?

79. If it takes a certain amount of money 3.7 years to double, at what annualinterest rate was the money invested?

80. AGRICULTURE The percentage offarmers in the United Statesworkforce has declined since theturn of the century. The percentof farmers in the workforce, f, canbe modeled by the function f(t) = 29e−0.036t, where t is time in years and t = 0 in 1920. Findthe percent of farmers in theworkforce in 1995. [Source:Bureau of Labor Statistics]

Solve each inequality, and graph the solution on a number line.

(LESSON 1.8)

81. |−3x| ≥ 15 82. |3x − 4| ≥ 12 83. ��3x−+4

2�� ≥ 5

Graph each function. (LESSON 2.7)

84. f(x) = �12�|x| 85. g(x) = −�x� 86. h(x) = [x − 3]

Graph each system. (LESSON 3.4)

87. � 88. � 89. �Factor each expression. (LESSON 5.3)

90. x2 − 3x − 10 91. 3x2 − 6x + 3 92. x2 − 49

93 Solve ln x + ln(x + 2) = 5 by graphing y = ln x + ln(x + 2) and y = 5 andfinding the x-coordinate of the point of intersection.

y + 2x ≥ 04 − 2x > yx ≤ 3

1 − 3x > y + 42x + 3y ≤ 8

y > 2x − 14 − 3x ≥ y

Look Back

Look Beyond

402 CHAPTER 6

Objectives

● Solve logarithmic andexponential equationsby using algebra andgraphs.

● Model and solve real-world problemsinvolving exponentialand logarithmicrelationships.

Solving Equationsand Modeling


GEOLOGY

On the Richter scale, the magnitude, M, of an earthquake depends on theamount of energy, E, released by the earthquake as follows:

M = �23� log �

10E11.8�

The amount of energy, measured in ergs, is based on the amount of groundmotion recorded by a seismograph at a known distance from the epicenter of the quake.

The logarithmic function for the Richter scale assigns very large numbers for the amount of energy, E, to numbers that range from 1 to 9. A rating of2 on the Richter scale indicates the smallest tremor that can be detected.Destructive earthquakes are those rated greater than 6 on the Richter scale.

8–9 near total damage 0.2

7.0–7.9 serious damage to buildings 14

6.0–6.9 moderate damage to buildings 185

5.0–5.9 slight damage to buildings 1000

4.0–4.9 felt by most people 2800

3.0–3.9 felt by some people 26,000

2.0–2.9 not felt but recorded 800,000

RICHTER SCALE RATINGS

MagnitudeApproximate number of

occurrences per yearResult near the epicenter

Physicists, chemists, andgeologists use exponential andlogarithmic equations to modelvarious phenomena, such as themagnitude of earthquakes.

WhyWhy

403LESSON 6.7 SOLVING EQUATIONS AND MODELING

One of the strongest earthquakes inrecent history occurred in MexicoCity in 1985 and measured 8.1 onthe Richter scale.

Find the amount of energy, E,released by this earthquake.

SOLUTION

Use a formula.

M = �23� log �10

E11.8�

8.1 = �23� log �10

E11.8� Substitute 8.1 for the magnitude, M.

12.15 = log �10E11.8�

1012.15 = �10E11.8� Use the definition of logarithm.

1011.8 • 1012.15 = E8.91 × 1023 ≈ E Write the answer in scientific notation.

The amount of energy, E, released by this earthquake was approximately 8.91 × 1023 ergs. In physics, an erg is a unit of work or energy.

To solve the logarithmic equation in Example 1, you must use the definition of a logarithm. However, solving exponential and logarithmic equations oftenrequires a variety of the definitions and properties from this chapter. Asummary of the definitions and properties that you have learned is given below.

Show how to solve M = �23

� log �10

E11.8� for E.

Use the properties of exponents and logarithms to show that loga (�1x�) = log �1a� x.

E X A M P L E 1

Exponential and Logarithmic Definitions and Properties

Definition of logarithm y = logb x if and only if by = x

Product Property logb mn = logb m + logb n

Quotient Property logb(�mn

�) = logb m − logb n

Power Property logb mp = p logb m

Exponential-Logarithmic b logbx = x for x > 0 Inverse Properties logb bx = x for all x

One-to-One Property If bx = by, then x = y.of Exponents

One-to-One Property If logb x = logb y, then x = y.of Logarithms

Change-of-base formula logc a = �l

l

o

o

g

gb

b

a

c�

SUMMARY

CHECKPOINT ✔

CRITICAL THINKING

PROBLEM SOLVING


GEOLOGY

Copy these propertiesand definitions into yournotebook for reference.

A seismogram produced by a seismograph

404 CHAPTER 6

Solve log x + log(x − 3) = 1 for x.

SOLUTION

Method 1 Use algebra.log x + log(x − 3) = 1

log[x(x − 3)] = 1 Apply the Product Property of Logarithms.x(x − 3) = 101 Write the equivalent exponential equation.

x2 − 3x − 10 = 0(x − 5)(x + 2) = 0

x = 5 or x = −2

CHECK

Let x = 5. Let x = −2.log x + log(x − 3) = 1 log x + log(x − 3) = 1

log 5 + log 2 =? 1 log(−2) + log(−5) = 1 Undefined1 = 1 True

Since the domain of a logarithmic function excludes negative numbers, theonly solution is 5.

Method 2 Use a graph.Graph y = log x + log(x − 3) and y = 1, and findthe point of intersection.

The coordinates of the point of intersection are(5, 1), so the solution is 5.

Solve log(x + 48) + log x = 2 by using algebra and a graph.

Solve 4e3x−5 = 72 for x.

SOLUTION

Method 1 Use algebra.4e3x−5 = 72

e3x−5 = 18

ln e3x−5 = ln 18 Take the natural logarithm of each side.

3x − 5 = ln 18 Use Exponential-Logarithmic Inverse Properties.

x = �ln 18

3+ 5

� Exact solution

x ≈ 2.63 Approximate solution

Method 2 Use a graph.Graph y = 4e3x−5 and y = 72, and find the pointof intersection.

The coordinates of the point of intersection areapproximately (2.63, 72), so the solution isapproximately 2.63.

E X A M P L E 2

TRY THIS

E X A M P L E 3

TECHNOLOGYGRAPHICS

CALCULATOR


TECHNOLOGYGRAPHICS

CALCULATOR



Solving Exponential Inequalities


1. Graph y1 = log x + log(x + 21) and y2 = 2 on the same screen.

2. For what value(s) of x is y1 = y2? y1 < y2? y1 > y2?

3. Explain how you can use a graph to solve log x + log(x + 21) > 2.

4. Graph y1 = 2e4x−1 and y2 = 38 on the same screen.

5. For what approximate value(s) of x is y1 = y2? y1 < y2? y1 > y2?

6. Explain how you can use a graph to solve 2e4x−1 < 38.

Newton’s Law of Cooling

An object that is hotter than its surroundings will cool off, and an object thatis cooler than its surroundings will warm up. Newton’s law of cooling statesthat the temperature difference between an object and its surroundingsdecreases exponentially as a function of time according to the following:

T(t) = Ts + (T0 − Ts)e −kt

T0 is the initial temperature of the object, Ts is the temperature of the object’ssurroundings (assumed to be constant), t is the time, and −k represents theconstant rate of decrease in the temperature difference (T0 − Ts).

When a container of milk is taken out of the refrigerator, its temperature is40˚F. An hour later, its temperature is 50˚F. Assume that the temperature of theair is a constant 70˚F.

a. Write the function for the temperature of this container of milk as afunction of time, t.

b. What is the temperature of the milk after 2 hours?c. After how many hours is the temperature of the milk 65˚F?

SOLUTION

a. First substitute 40 for T0 and 70 for Ts, and simplify.

T(t) = Ts + (T0 − Ts)e−kt

T(t) = 70 + (40 − 70)e−kt

T(t) = 70 + (−30)e−kt

Since T(1) = 50, substitute 1 for t and 50 for T(t), and solve for −k.

50 = 70 − 30e−k

30e−k = 20e−k = �

23�

ln e−k = ln �23�

−k = ln �23�

CHECKPOINT ✔

CHECKPOINT ✔

E X A M P L E 4


PHYSICS

TECHNOLOGYGRAPHICS

CALCULATOR


Activities OnlineGo To: go.hrw.comKeyword:MB1 Spacecraft

406 CHAPTER 6

Substitute ln �23� for −k and simplify to get the function for the temperature

of this container of milk.

T(t) = 70 − 30e−kt

T(t) = 70 − 30e(ln �23�)t

T(t) = 70 − 30(�23�)

tApply the Exponential-Logarithmic Inverse Property.

The function for the temperature of this container of milk is

T(t) = 70 − 30(�23�)

t.

b. Find T(2).

T(t) = 70 − 30(�23�)

t

T(2) = 70 − 30(�23�)

2≈ 56.7

The temperature of the milk after2 hours is approximately 56.7˚F.

c. Substitute 65 for T(t), and solve for t.

T(t) = 70 − 30(�23�)

t

65 = 70 − 30(�23�)

t

30(�23�)

t= 5

(�23�)

t= �

16�

t ln �23� = ln �

16�

t = ≈ 4.42

It will take approximately 4.42 hours, or about 4 hours and 25 minutes, forthe milk to warm up to 65˚F.

1. Explain how to solve the exponential equation ex+7 = 98 by algebraicmethods.

2. How can you solve the logarithmic equation log2 x + log2(x + 3) = 2 byalgebraic methods?

3. Explain how to solve exponential and logarithmic equations by graphing.

ln �16�

�ln �

23�

ExercisesExercises

Communicate

TECHNOLOGYGRAPHICS

CALCULATOR



Solve each equation for x. Write the exact solution and the approximate

solution to the nearest hundredth, when appropriate.

8. 3x = 34 9. 32x = 81 10. 5x −2 = 25

11. x = log3 �217�

12. x = log4 �614�

13. logx �116�

= −2

14. 4 = logx �116�

15. e2x = 20 16. e−2(x+1) = 2

17. ln(2x − 3) = ln 21 18. ln(x + 3) = 2 ln 4 19. 102x + 75 = 150

20. e−4x − 22 = 56 21. 3 ln x = ln 4 + ln 2 22. ln x + ln(x + 1) = ln 2

23. 2 ln x + 2 = 1 24. 3 ln x + 3 = 1 25. 3 log x + 7 = 5

Solve each equation for x. Write the exact solution and the approximate

solution to the nearest hundredth, when appropriate.

26. ln(3�x�) = �ln� x� 27. log x3 = (log x)3

28. GEOLOGY On May 10, 1997, a light earthquake with a magnitude of4.7 struck the Calaveras Fault 10 miles east of San Jose, California. Findthe amount of energy, E, released by this earthquake.

29. GEOLOGY In 1976, an earthquake that released about 8 × 1019 ergs ofenergy occurred in San Salvador, El Salvador. Find the magnitude, M, ofthis earthquake to the nearest tenth.

30. PHYSICS A hot coal (at a temperature of160˚C) is immersed in ice water (at atemperature of 0˚C). After 30 seconds, thetemperature of the coal is 60˚C. Assumethat the ice water is kept at a constanttemperature of 0˚C.a. Write the function for the temperature,

T, of this object as a function of time, t,in seconds.

b. What will be the temperature of thiscoal after 2 minutes (120 seconds)?

c. After how many minutes will thetemperature of the coal be 1˚C?


4. GEOLOGY In 1989, an earthquake that measured 7.1 on the Richter scaleoccurred in San Francisco, California. Find the amount of energy, E,released by this earthquake. (EXAMPLE 1)

5. Solve log(x − 90) + log x = 3 for x. (EXAMPLE 2)

6. Solve 0.5e0.08t = 40 for x. (EXAMPLE 3)

7. PHYSICS When the air temperature is a constant 70˚F, an object cools from170˚F to 140˚F in one-half hour. (EXAMPLE 4)

a. Write the function for the temperature of this object, T, as a function oftime, t.

b. What is the temperature of this object after 1 hour?c. After how many hours is the temperature of this object 90˚F?

Practice and Apply

C H A L L E N G E


Cooling Temperature

of a Hot Coal

Time (s)

160

140

120

100

80

60

40

20

00 15 30 45

Te

mp

era

ture

(˚C

)




408 CHAPTER 6

31. GEOLOGY Compare the amounts of energy released by earthquakes that differby 1 in magnitude. In other words, how much more energy is released by anearthquake of magnitude 6.8 than an earthquake of magnitude 5.8?

32. PSYCHOLOGY Educational psychologists sometimes use mathematicalmodels of memory. Suppose that some students take a chemistry test.After a time, t (in months), without a review of the material, they take an equivalent form of the same test. The mathematical model a(t) = 82 − 12 log(t + 1), where a is the average score at time t, is afunction that describes the students’ retention of the material.a. What is the average score when the students first took the test (t = 0)?b. What is the average score after 6 months?c. After how many months is the average score 60?

33. BIOLOGY A population of bacteria grows exponentially. A population thatinitially consists of 10,000 bacteria grows to 25,000 bacteria after 2 hours.a. Use the exponential growth function, P(t) = P0ekt, to find the value

of k. Then write a function for this population of bacteria in terms of time, t. Round the value of k to the nearest hundredth.

b. How many bacteria will the population consist of after 12 hours,rounded to the nearest hundred thousand?

c. How many bacteria will the population consist of after 24 hours?

34. DEMOGRAPHICS The population of India was estimated to be 574,220,000in 1974 and 746,388,000 in 1984. Assume that this population growth isexponential. Let t = 0 represent 1974 and t = 10 represent 1984.a. Use the exponential growth function, P(t) = P0ekt, to find the value of k.

Then write the function for this population as a function of time, t.b. Estimate the population in 2004, rounded to the nearest hundred

thousand.c. Use the function you wrote in part a to estimate the year in which the

population will reach 1.5 billion.

35. ARCHEOLOGY Refer to the discussion of radioactive decay on page 395.Suppose that an animal bone is unearthed and it is determined that theamount of carbon-14 it contains is 40% of the original amount.a. Use the decay function for carbon-14, N(t) = N0e −0.00012t, to write an

equation using the percentage of carbon-14 given above.b. Use the equation you wrote in part a to find the approximate age, t, of

the bone.


The map shows the locations of the threestrongest earthquakes in the United Statesin 1995.

6.8 magnitudeearthquake off thecoast of northernCalifornia onFebruary 19, 1995

5.8 magnitude earthquakein Ridgecrest, California,northeast of Los Angeleson September 20, 1995

5.7 magnitude earthquakein Brewster County,Texas, on April 14, 1995

A C T I V

I TY

PO

RTFOLIO

PortfolioExtensionGo To: go.hrw.comKeyword:MB1 Newton

This activity requires the data collected for the PortfolioActivities on pages 361, 369,and 384.

1. Refer to your data from the PortfolioActivity on page 361.a. Use Newton’s law of cooling, found in

Example 4 on page 405, to write a functionthat models the temperature of the probeas it cooled to the temperature of ice water.

b. Compare the graph of this function withthe graph of the exponential function thatyou generated for the same data in thePortfolio Activity on page 369. (Hint: Youcan use the table function on your graphicscalculator to compare the y-values of thesefunctions with the original values.)

2. a. Repeat part a of Step 1, using the datacollected in the Portfolio Activity on page 384.

b. Repeat part b from Step 1, comparingyour new graph with the graphs of boththe exponential function and theapproximating function from thePortfolio Activity on page 384.

WORKING ON THE PROJECTYou should now be able to complete theChapter Project.

Graph each system of linear inequalities. (LESSON 4.8)

36. � 37. � 38. �Solve each equation for x. (LESSON 5.2)

39. x2 − 3 = 46 40. 7 − x2 = 4

Solve for x, and check your answers. (LESSON 6.4)

41. logb(x2 − 11) = logb(2x + 4) 42. log10(8x + 1) = log10(x2 − 8)

43. loga(x2 + 1) + 2 loga 4 = loga 40x 44. 2 logb x − logb 3 = logb(2x − 3)

INVESTMENTS Assume that all interest rates are compounded

continuously in Exercises 45–47. (LESSON 6.6)

45. How long will it take an investment of $5000 to double if the annualinterest rate is 5%?

46. How long will it take an investment to double at 8% annual interest?

47. If it takes a certain amount of money 3.2 years to double, at what annualinterest rate was the money invested?

48. Graph each function and compare the shapes of the graphs.a. y = x2 b. y = x3 − 2x c. y = x4 − 2x2

−y + 3 ≤ 12−x < y + 8

−x ≤ −3y − 5 > −3

2x − 5y < 4−3x ≥ 2y

Look Back

Look Beyond


LESSON 6.7 SOLVING EQUATIONS AND MODELING 409

412 CHAPTER 6

Chapter Review and Assessment

Write and evaluate exponential expressions.

The world population rose to about 5,734,000,000in 1995. The world population was increasing atan annual rate of 1.6%. Write and evaluate anexpression to predict the world population in2020. [Source: Worldbook Encyclopedia]

5,734,000,000(1.016)x

5,734,000,000(1.016)25 ≈ 8,527,000,000

The projected world population for 2020 is about8.5 billion people.

1. INVESTMENTS The value of a painting is$12,000 in 1990 and increases by 8% ofits value each year. Write and evaluate anexpression to estimate the painting’s value in 2005.

2. DEPRECIATION The value of a new car is$23,000 in 1998; it loses 15% of its value eachyear. Write and evaluate an expression toestimate the car’s value in 2005.

Key Skills Exercises

LESSON 6.1

Key Skills & Exercises

asymptote . . . . . . . . . . . . . . . 362base . . . . . . . . . . . . . . . . . . . . 362change-of-base formula . . . 388common logarithm . . . . . . . . 385compound interest formula . . 365continuous compounding

formula . . . . . . . . . . . . . . . . 393effective yield . . . . . . . . . . . . 365exponential decay . . . . . . . . 363exponential expression . . . 355

exponential function . . . . . . 362exponential growth . . . . . . . 363Exponential-Logarithmic

Inverse Properties . . . . . . 380logarithmic function . . . . . . . 372multiplier . . . . . . . . . . . . . . . . 355natural base . . . . . . . . . . . . . 393natural exponential

function . . . . . . . . . . . . . . . 393natural logarithmic

function . . . . . . . . . . . . . . . 394

Newton’s law of cooling . . . 405One-to-One Property of

Exponents . . . . . . . . . . . . . 372One-to-One Property of

Logarithms . . . . . . . . . . . . . 380Power Property of

Logarithms . . . . . . . . . . . . . 379Product Property

of Logarithms . . . . . . . . . . 378Quotient Property

of Logarithms . . . . . . . . . . 378

Classify an exponential function as

exponential growth or exponential decay.

When b > 1, the function f(x) = bx representsexponential growth.

When 0 < b < 1, the function f(x) = bx representsexponential decay.

Calculate the growth of investments.

The total amount of an investment, A, earning

compound interest is A(t) = P(1 + �nr

�)nt

, where P is

the principal, r is the annual interest rate, n is thenumber of times interest is compounded per year,and t is the time in years.

Identify each function as representing

exponential growth or decay.

3. f(x) = 4(0.89)x 4. g(x) = �13�(1.06)x

5. h(x) = 5(1.06)x 6. j(x) = 25(�25�)x

INVESTMENTS For each compounding periodbelow, find the final amount of a $2400investment after 12 years at an annual interestrate of 4.5%.

7. annually 8. quarterly

9. daily


LESSON 6.2

VOCABULARY

413CHAPTER 6 REVIEW

Use the common logarithmic function to

solve exponential and logarithmic equations.

4 = log x 2x = 34104 = x log 2x = log 34

10,000 = x x log 2 = log 34

x = �lloogg

324

�

x ≈ 5.09

Apply the change-of-base formula to evaluate

logarithmic expressions.

The change-of-base formula is loga x = �lloo

gg

b

b

ax

�,

where a ≠ 1, b ≠ 1, and x > 0.

log2 5 = �lloo

gg

52� ≈ 2.32

Solve each equation. Give your answers to

the nearest hundredth.

29. log x = 8 30. log 0.01 = x − 5

31. 5x + 100 = 98 32. 5 − 2x = 40

33. 7 + 32x −1 = 154 34. 3 + 73x +1 = 346

Evaluate each logarithmic expression to the

nearest hundredth.

35. log3 14 36. log16 3

37. log0.5 6 38. log1.5 10

39. CHEMISTRY What is [H+] of a carbonated sodaif its pH is 2.5?


LESSON 6.5

Write equivalent forms of exponential and

logarithmic equations.

34 = 81 is log3 81 = 4 in logarithmic form.

log2 64 = 6 is 26 = 64 in exponential form.

Use the definitions of exponential and

logarithmic functions to solve equations.

v = log6 36 3 = log4 v 4 = logv 816v = 36 43 = v v4 = 816v = 62 64 = v v4 = 34

v = 2 v = 3

10. Write 52 = 25 in logarithmic form.

11. Write log3 27 = 3 in exponential form.

12. Write log3 �19� = −2 in exponential form.


13. v = log8 64 14. logv 4 = 2

15. 2 = log12 v 16. 3 = logv 1000

17. log2 v = −3 18. log27 3 = v

19. logv 49 = 2 20. log4 �116�

= v


LESSON 6.3

Use the Product, Quotient, and Power

Properties of Logarithms to simplify and

evaluate expressions involving logarithms.

Given log3 7 ≈ 1.7712, log3 63 can beapproximated as shown below.

log3 63 = log3 9 + log3 7= 2 + 1.7712 ≈ 3.7712

log5 257 = 7 log5 25 = 7 • 2 = 14

Given log7 5 ≈ 0.8271 and log7 9 ≈ 1.1292,

approximate the value of each logarithm.

21. log7 45 22. log7 �59� 23. log7 35

Write each expression as a single logarithm.

Then simplify, if possible.

24. log5 3 + log5 6 + log5 9

25. log 6 – log 3 + 2 log 7


26. 2log2 12 27. log7 73 28. log6 367


LESSON 6.4

Ch06 p412-421 5/21/03 7:57 AM Page 413

414

Evaluate exponential functions of base e and

natural logarithms.

Using a calculator and rounding to the nearestthousandth, e2.5 ≈ 12.182 and ln 3.5 ≈ 1.253.

Model exponential growth and decay

processes by using base e.

The continuous compounding formula is A = Pert,where A is the final amount when the principal Pis invested at an annual interest rate of r for t years.

Evaluate each expression to the nearest

thousandth.

40. e0.5 41. e−5

42. ln 5 43. ln 0.05

44. INVESTMENTS Sharon invests $2500 at anannual interest rate of 9%. How much is theinvestment worth after 10 years if the interestis compounded continuously?


LESSON 6.6

Solve logarithmic and exponential equations.

logx �312�

= −5 ln x3 + 5 = 1

x −5 = �312�

3 ln x = −4

x −5 = 2−5 ln x = −�43�

x = 2 x = e− �

43�

x ≈ 0.264

ln(x + 6) = 2 ln 3ln(x + 6) = ln 9

x + 6 = 9x = 3

Solve each equation for x. Write the exact

solution and the approximate solution to the

nearest hundredth, when appropriate.

45. logx �1128�

= −7

46. ln(2x) = 4 ln 2

47. x log �16� = log 6

48. ln �x� − 3 = 1

49. HEALTH The normal healing of a wound canbe modeled by A = A0e−0.35n, where A is thearea of the wound in square centimeters aftern days. After how many days is the area of thewound half of its original size, A0?


LESSON 6.7

Applications

50. BIOLOGY Given favorable living conditions, fruit flypopulations can grow at the astounding rate of 28% perday. If a laboratory selects a population of 25 fruit fliesto reproduce, about how big will the population be after3 days? after 5 days? after 1 week?

51. PHYSICS Suppose that the sound of busy traffic on afour-lane street is about 108.5 times the intensity of thethreshold of hearing, I0. Find the relative intensity, R, indecibels of the traffic on this street.

52. PHYSICS Radon is a radioactive gas that has a half-life ofabout 3.8 days. This means that only half of the originalamount of radon gas will be present after about 3.8 days.Using the exponential decay function A = Pe–kt, find thevalue of k to the nearest hundredth, and write thefunction for the amount of radon remaining after t days.

CHAPTER 6

Fruit fly

415CHAPTER 6 TEST

Chapter Test

1. DEMOGRAPHICS The population of Petoskey,Michigan, was 6076 in 1990 and was growing atthe rate of 3.7% per year. The city planners wantto know what the population will be in the year2025. Write and evaluate an expression to estimatethis population.

2. INCOME TAX The government allows for lineardepreciation of capital expenditures for income taxpurposes at the rate of 10% per year. What will bethe value of a $150,000 tool and die machine after7 years of use?

Tell whether each function represents

exponential growth or decay.

3. f(x) � 3.6(1.01)x

4. g(t) � 0.015(1.23)t

5. h(t) � �7�34��

58��

t

6. j(x) � 2500(0.25)x

INVESTMENTS For each compounding period

below, find the final amount of a $5000

investment after 10 years at a 5.6% annual

interest rate.

7. daily 8. monthly

9. quarterly 10. annually

Write each logarithmic equation in

exponential form and each exponential

equation in logarithmic form.

11. log381 � 4 12. 28 � 256

13. ��14��

−5� 1024 14. log5 � �4


15. 3 � logv343

16. log9729 � v

17. log6v � 5

Write each expression as a single logarithm.

Then simplify, if possible.

18. log25 − 3log23 + log26

19. log7�14� + 2log74 − �

12�log716


20. 5log5

32 21. log636

22. log99 23. logbb(x − 2)

Solve each equation. Give your answers to

the nearest hundredth.

24. log4x � 6.2 25. 3x + 2 � 238

26. 274 − 5x � 198 27. log6468 � x

28. SEISMOLOGY The amount of energy E, in ergs,released by an earthquake of magnitude M is givenby the formula E � 10 (1.5M � 11.8). What is thedifference in the amount of energy released by anearthquake of magnitude 6.5 and one ofmagnitude 8.7?

Evaluate each expression to the nearest

thousandth.

29. e3.4 30. lnπ

31. e−3.25 32. ln(e1.618)

33. ARCHAEOLOGY The age of an artifact can bedetermined using carbon-14 dating with theequation N(t) � N0e-0.00012t. What is theapproximate age of an artifact if a sample revealsthat it contains 34% of its original carbon-14?

Solve each equation for x. Write the exact

solution and the approximate solution to the

nearest hundredth, when appropriate.

34. 3x � 52.3 35. ln(x + 1) � 2 ln 4

36. logx + log(x + 3) � 1

1625

23

65. approximately 0 ≤ t < 2.48 67. about 2.32seconds 69. between 16 and 159 pairs, inclusive71.yes 73. yes 75. x = ±�8� 77. x = ±�18�79. −1 + 4i 81. 6 + i

CHAPTER REVIEW AND ASSESSMENT

1. f(x) = –x2 + 3x + 4; a = −1, b = 3, and c = 43. (1.5, 1.25) 5. opens down; maximum7. x = ±�8�; x ≈ ±2.83 9. x ≈ ±7 11. x = –5 or x = 11

13. x = –1 ± ��

754��; x ≈ –3.78 or x ≈ 1.78 15. c ≈ 6.4

17. a ≈ 9.7 19. a ≈ 24.5 21. b ≈ 0.7 23. 7x(x – 3)25. (x + 5)(x + 2) 27. (t + 3)(t – 8)29. (x + 2)(x – 10) 31. (x + 5)(x – 4)33. (3y + 2)(y – 1) 35. (4 + 3x)(4 – 3x) 37. (x – 8)2

39. x = 4 or x = 6 41. t = – �52� or t = �

23�

43. x = �15� ± �

�56�

45. x = −12 or x = 7

47. x = –4 or x = �12� 49. x = – �

43� or x = 2

51. y = –3(x + 1)2 – 4; (−1, −4)

53. y = 4�x – �98��2

– �41

96�

; ��98� , – �

41

96�� 55. x = – �

35� ± �

�1706�

�

57. x = –4 or x = –2 59. x = �112�

± ��1723�

�

61. x = – �12� ± �

�25�� 63. ��

12�, – �

449�� 65. (−6, −31)

67. 2 real solutions 69. no real solutions

71. x = −5 − 3i or x = −5 + 3i 73. x = 3 ± �i�

28�

�

75. x = �13� ± �

i�620�� 77. 1 − 7i 79. −1 81. 6 − 9i

83. 2 − 4i

85. �10� 87. �2�

89. f(x) = 5x2 + 2x – 9 91. f(x) = –5x2 – x + 693. f(x) = 3x2 – 3x – 3 95. f(x) = 7x2 – 197. f(x) ≈ –0.7x2 + 1.5x – 3.199. x < 2 or x > 6

101. x ≤ –5 or x ≥ –2

103. x < – �34� or x > 3

105. 0 ≤ x ≤ �52�

107. 109.

111.

113. about 2.8 seconds

Chapter 6

LESSON 6.1

TRY THIS (p. 356)188,700,000; 194,400,000

TRY THIS (p. 357)192 milligrams, 62.9 milligrams

Exercises5. 1.055 6. 1.0025 7. 0.97 8. 0.995 9. 8 10. 135011. 0.512 12. 42.1875 13. 31,400,000 14. 31.0milligrams and 27.3 milligrams 15. 1.07 17. 0.9419. 1.065 21. 0.9995 23. 1.00075 25. 1.927. 1,638,400 29. 37.9 31. 394.0 33. 941,013.735. 32.6 37a. 8000 bacteria b. 32,000 bacteria39a. 1200 bacteria b. 4800 bacteria 41a. 6975bacteria b. 62,775 bacteria 43. exponential45. linear 47. 278,700,000 49. 1,359,600,000 and1,399,800,000 51a. 1,002,600,000 and 1,140,800,000b. 13.79% c. 1,107,500,000

53. 310,000 gallons 55. 2 57. 7 59. �nm

1

3

5� 61.

63. reflection across y-axis and horizontalcompression by a factor of �

12� 65. reflection across

x-axis, horizontal stretch by a factor of 2, and verticaltranslation 3 units up 67. reflection across x-axis,vertical stretch by a factor of 5, horizontal translation2 units to the right, and vertical translation 4 unitsdown 69. opens down; maximum value

LESSON 6.2

TRY THIS (p. 364) TRY THIS (p. 366)a. exponential growth; �

13� 7.2%

b. exponential decay; �14�

1�12xy 2

x

y

2010−10

−20

−10−20

x

y

42−2

−4

−2−4

4

2

x

y

84–4

–8

–4–8

8

4

–6 –4–5 –3 –2 –1 0 1 2 3 654

52

–4 –2–3 –1

–

0 1 2 3 4 5 6 7 8

34

–4 –2–3–6 –5 –1 0 1 2 3 4 5 6

–4 –2–3 –1 0 1 2 3 4 5 6 7 8

1

Imaginary

1Real

1 + i

√2

–6

–4

1

Imaginary

–3 – i–6

Real

√10

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Exercises5. exponential decay; 1 6. exponential growth; 37. exponential decay; 5 8. $334.56, $336.71, and$337.46 9. 4.7% 11. quadratic 13. exponential15. quadratic 17. growth 19. decay 21. decay23. decay 25. d 27. b 29. $3207.14 31. $2013.8033. $2132.45 35a. h(x) = 8x; f(x) = 2x b. 1c. Both functions have a domain of all real numbersand a range of all positive real numbers. 37. when a = 0 or when b = 1 39. g is f compressed verticallyby a factor of �

12�. 41. g is f stretched vertically by a

factor of 2 and translated vertically 3 units down.43. g is f translated 1 unit to the right and stretchedvertically by a factor of 5. 45. g is f stretchedvertically by a factor of 2. 47. 3.8% 49. 1.3%51. $2615.16 53a. Final amount is doubled:$114,674. b. Final amount is more than 10 timeslarger: $586,954.26. c. Final amount is more than 11 times larger: $657,506.29. d. doubling theinvestment period 55. {(2, 7), (–1, 3), (2, 2), (0, 0)};

not a function 57. y = ± �33x�

; not a function

59. y = ±�–�x�; not a function

61.

63. does not exist 65. 67. 69. y = 3x2 – 2x + 4

LESSON 6.3

TRY THIS (p. 371, Ex. 1)

TRY THIS (p. 371, Ex. 2) TRY THIS (p. 373, Ex. 3)−2.037 a. v = 3 b. v = 5

c. v = 729

TRY THIS (p. 373, Ex. 4)[H+] ≈ 0.00018 moles per liter

Exercises4. log4 16 = 2 5. 52 = 25 6. 2.754 7. −2.699 8. 29. 12 10. 16 11. 10−5, or 0.0000113. log5 625 = 4 15. log6 216 = 3 17. log7 �4

19�

= –2

19. log16 2 = �14� 21. log�

19

� 81 = –2 23. log�12

� �18� = 3

25. 103 = 1000 27. 10−1 = 0.1 29. 73 = 343

31. 2−5 = �312�

33. 3−3 = �217�

35. 144�12

�= 12 37. 2 39. 4

41. 0 43. −4 45. 1.08 47. 3.55 49. 0.28 51. −1.4653. −3.04 55. −1.35 57. 3 59. 2 61. 1 63. −2

65. −2 67. 0 69. 49 71. 3 73. 2 75. �614�

77. 1 79. 581. 64 83. 2 85. 3

87.

Tables may vary. Sample tables provided.

89. 4.5 91. stretched vertically by a factor of 393. compressed vertically by a factor of �

12� and

translated 1 unit up 95. reflected across the x-axisand translated 2 units to the right 97. 10−10 molesper liter 99. 3.98 × 10−8 moles per liter 101. 5261.1feet 103. Identity Property of Multiplication105. Commutative Property of Multiplication107. Inverse Property of Multiplication

109. A−1 ≈ 111. ±i

LESSON 6.4

TRY THIS (p. 379, Ex. 1) TRY THIS (p. 379, Ex. 2)

a. 4.17 b. −0.4150 a. log4 3 b. logb �43x�

TRY THIS (p. 380, Ex. 3) TRY THIS (p. 380, Ex. 4)300 a. 7 b. 13

Exercises5. 3.5424 6. −0.7712 7. log3 �

xyz� 8. log2 �

95� 9. 16

0.364 0.2730.091 −0.182

x

y

4–2

–4

–2–4

4y = f(x)

y = f –1(x)

–32 14–27 12

–2 40 –3851 52 –9

x

y

8 10642

2

6

2

4

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Exponential form Logarithmic form

25 = 32 log2 32 = 5

103 = 1000 log10 1000 = 3

3–2 = 19

log3 19

= –2

16�12

�= 4 log16 4 = 1

2

x f(x) = 3–x

–3 27

–2 9

–1 3

0 1

1 �13�

2 �19�

3 �217�

x f –1(x) = log �13� x

27 –3

9 –2

3 –1

1 0

�13� 1

�19� 2

�217�

3

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10. 12 11. 3 12. x = 4 13. log8 5 + 115. log3 x – 2 17. 1.9535 19. 4.8074 21. 2.953523. 2.9191 25. −0.3685 27. 0.1610 29. log2 35

31. log3 5 33. log2 �2x

� 35. log7 2y 37. log2 �mn2

5�

39. logb 41. log7 �x7

2� 43. 8 45. 5 47. 12 49. 8

51. 5 53. −5 55. x = 3 or x = 4 57. x = 8 59. x = 361. x = 3 63. always 65. always 67. never69. always 71. sometimes 73. x = 24375. 1.83 times greater 77. 12

79. = 81.

83. 85. $802.35; $866.99; $936.49

LESSON 6.5

TRY THIS (p. 387) TRY THIS (p. 388)x ≈ 3.21 1.72

Exercises4. 18.75 5. 1013 times louder 6. 0.67 7. 3.08 8. 5.529. 0.43 11. 5.61 13. 2.64 15. 2.05 17. −1.2319. 1.96 21. 0.16 23. 10 25. 4.81 27. 3.66 29. 2.4931. 2.46 33. 0.43 35. 0.53 37. 0.16 39. −341. −2.01 43. −0.89 45. 7 47. 105 decibels49. 1011.5 times louder 51. 30 decibels 53. 1012.8

times louder 55. 10−14 < [H+] < 10−7 57. ≈1.359. ≈10.5 61. (5, 2) 63. (−3, 2)65. g(x) = 2x2 – 8x – 10 67. f(x) = –3x2 – 2x + 169. −5 or 3 71. exponential growth 73. exponentialgrowth

LESSON 6.6

TRY THIS (p. 393) TRY THIS (p. 394)403.429; 0.717 $716.66

TRY THIS (p. 395)about 15.26 years

Exercises5. 20.086 6. 33.115 7. $2250.88; $2260.23 8. 1.6099. 0.916 10. about 9.24 years 11. about 20,066 yearsold 13. 8103.084 15. 29.964 17. 3.15419. 2.320 21. 1.284 23. 1.946 25. 11.513

27. −0.006 29. 0.805 31. undefined 33. ln �12�, ln 1,

e0, e 35. log 1.3, ln 1.3, e1.3, 101.3 37. always true39. sometimes true 41. 5 43. 25 45. 4 47. 849. ln 1 = x 51. 5 ≈ e1.61 53. ln 1.99 ≈ 0.6955. 7.93 57. 0.42 59. −9.97 61. vertical stretch by a factor of 6 and vertical translation 1 unit up

63. horizontal translation 1 unit to the left,horizontal compression by a factor of �

14�, and vertical

compression by a factor of 0.25 65. vertical stretchby a factor of 3 and vertical translation 1 unit to the

left 67. horizontal compression by a factor of �15�,

vertical compression by a factor of 0.5, and verticaltranslation of 2 units down 69. f to h: reflectionacross the y-axis and horizontal compression by a

factor of �12�; g to h: reflection across the y-axis; i to h:

horizontal compression by a factor of �12� 71. The

only changes are the following: a. no changesb. x-intercept: changes c. x-intercept: changesd. domain: real numbers greater than the value ofthe translation; asymptotes: x is the value of thetranslation; x-intercept: value of the translation plus1 73a. $285,000,000 b. during 1999 75. No;according to the tests, the chest is only about 700 yearsold. 77. almost 11 years and 7 months 79. 18.7%81. x ≥ 5 or x ≤ −5

83. x ≥ 6 or x ≤ – �232�

85. 87.

89. 91. 3(x – 1)2

LESSON 6.7

TRY THIS (p. 404)Method 1

log(x + 48) + log x = 2log[x(x + 48)] = 2

x(x + 48) = 102

x2 + 48x – 100 = 0(x + 50)(x – 2) = 0

x = −50 or x = 2

–2

–4

4

y

4–2–4x

–4

4

2

y

42–2–4x

–2

–4

4

2

y

42–2–4x

–12 –8–10 –6 –4 –2 0 2 4 6 12108

– 223

–6 –4–5 –3 –2 –1 0 1 2 3 654

0.5 0.3 0 2.20 −8.5 1.2 −24.4

1.3 0 3.3 29

−3 2 114 1 5

7−12−5

xyz

3 2 −15 3 −22 3 −1

m4n�12

�

�8p3

Check: Let x = −50.log(x + 48) + log x = 2log(–2) + log(–50) = 2 Undefined

Let x = 2.log(x + 48) + log x = 2

log 50 + log 2 = 22 = 2 True

The solution is 2.Method 2Graph y = log(x + 48) + log x and y = 2, and find thepoint of intersection. The solution is 2.

Exercises4. 2.82 × 1022 ergs 5. x = 100 6. x ≈ 54.787a. T(t) = 70 + 100e−0.7133t, or T(t) = 70 + 100(0.7)2t

b. 119˚F c. about 2.26 hours, or about 2 hours and

16 minutes 9. 2 11. −3 13. 4 15. �ln

220� ≈ 1.50

17. 12 19. �log

275� ≈ 0.94 21. 2 23. e

− �12

�≈ 0.61

25. 10− �

23

�≈ 0.22 27. 1; 10− ≈ 0.02; 10 ≈ 53.96

29. 5.4 31. about 31.6 times 33a. P(t) = 10,000e0.46t

b. 2,500,000 bacteria c. 623,000,000 bacteria35a. 0.40N0 = N0e−0.00012t b. 7636 years old37.

39. x = 7 or x = –7 41. x = 5 43. x = �12� and x = 2

45. ≈13.86 years 47. 21.7%

CHAPTER REVIEW AND ASSESSMENT

1. 12,000(1.08)t; about $38,066 3. exponential decay5. exponential growth 7. $4070.12 9. $4118.28

11. 33 = 27 13. 2 15. 144 17. �18� 19. 7 21. 1.9563

23. 1.8271 25. log 98 27. 3 29. 100,000,000.0031. no solution 33. 2.77 35. 2.40 37. −2.5839. 10−2.5 41. 0.007 43. −2.996 45. 2 47. −149. about 2 days 51. 85 decibels

Chapter 7

LESSON 7.1

TRY THIS (p. 425) TRY THIS (p. 425, Ex. 2)a. cubic polynomial 17.6875b. quintic trinomial

TRY THIS (p. 425, Ex. 3) TRY THIS (p. 427)4x3 – 7x2 + 13x − 12 3x3 – 11x2 – 10x – 7

TRY THIS (p. 428)a. The graph of this cubic function is S-shaped andhas 2 turns.b. The graph of this quartic function is W-shapedand has 3 turns.

Exercises4. quartic trinomial 5. quintic polynomial 6. 157. 2x3 + 3x + 7 8. 3x3 – 3x2 + 7x + 5 9. S-shapedwith 2 turns 10. W-shaped with 3 turns11. 5x3 + 2x2 + 4x + 1 13. 3.3x8 + 2.7x3 + 4.1x2

15. �x7

9� + �1

x3

7� – �

23� 17. yes; quintic polynomial 19. no

21. yes; quartic trinomial 23. no 25. yes; trinomialof degree 6 27. no 29. −17 31. −138 33. 414

35. 8�58� 37. −39.625

39. 3x3 + 4x2 + 2x + 4; cubic polynomial41. –2x4 – 4x3 + 10x2 – 5x + 1; quartic polynomial43. 4x3 + 8x2 – 7x + 8; cubic polynomial45. –2x3 + 5x2 + 4; cubic trinomial

47. �23�x3 +�

13�x2 + x + �

53�; cubic polynomial

49. 2.5x4 + 7.6x3 – 3.2x2 + 7.8x; quartic polynomial51. S-shaped with 2 turns 53. W-shaped with 3 turns55. S-shaped with 2 turns 57. W-shaped with 3 turns59. a = −2, b = 4, c = 7, d = −7 61. 28x2 + 54x

63. $23,996,445 65. 67. 69. 71. 73. x = –3 or x = 5

75. x = –9 or x = –8 77. x = –2 or x = – �13�

79. x = �–1 –

5i�19�� or x = �

–1 +5i�19��

LESSON 7.2

TRY THIS (p. 434)a. maximum of 6.6

minimum of 3.9b. decreases for all values of x except over the

interval of approximately –0.2 < x < 1.5, where itincreases

TRY THIS (p. 436)a. falls on the left and rises on the rightb. rises on the left and the right

TRY THIS (p. 437)y ≈ –0.10x4 + 2.74x3 – 26.16x2 + 100.55x – 112.79

Exercises5. maximum of 2.1, minimum of −0.6; increases forall values of x except over the interval ofapproximately –1.2 < x < 0.5, where it decreases6. rises on the left and the right 7. rises on the leftand falls on the right

–4 75 –3

–1 95 1

–2 810 6

–2 55 1

4 62–2x

–2

6

4

y

�3��3�

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ms. parsons math class - background: prehistoric rock art...

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