8 logarithmic functions exponential and - tench's...

Chapter 8 l Exponential and Logarithmic Functions 307

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8.1 Logs, Exponents, and MoreSolving Exponential and Logarithmic

Equations | p. 309

8.2 Decibels, pH, and the Richter ScaleLogarithms and Problem Solving

Part I | p. 313

8.3 Depreciation, Population Growth, and Radioactive DecayLogarithms and Problem Solving

Part II | p. 319

8.4 Money, Money, Money!Loans and Investments | p. 327

Exponential and Logarithmic Functions8

CHAPTER

Scarlet macaws are native to the jungles of Southern Mexico and Central America, and can live

up to 75 years. However, macaws and other birds are threatened by deforestation, which destroys

their native habitat. You will use logarithmic functions to calculate the number of parrots in a

shrinking rain forest.

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308 Chapter 8 l Exponential and Logarithmic Functions

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Lesson 8.1 l Solving Exponential and Logarithmic Equations 309

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8.1 Logs, Exponents, and MoreSolving Exponential and Logarithmic Equations

Problem 1 Solving for the Result or Inverse Log

Solve each logarithmic equation by first converting to an exponential equation.

1. log2 x � 5 2. log x � 5

3. In x � 5 4. log3 x � �3

5. log5 x � 0 6. log

6 x � 2.34

ObjectiveIn this lesson you will:

l Solve exponential and logarithmic equations.

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7. log x � 5.34 8. ln x � 1.202

Solve each logarithmic equation by first converting to an exponential equation.

1. log2 4 � x 2. log 4 � x

3. ln 4 � x 4. log2 1 ___ 16

� x

5. log7 343 � x

Problem 2 Solving for the Log or Exponent

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Lesson 8.1 l Solving Exponential and Logarithmic Equations 311

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Problem 3 Solving for BaseSolve each logarithmic equation by converting to an exponential equation and

applying the properties of exponents.

1. logx 4 � 2 2. log

x 5 � 1 __

2

3. logx 4 � 3 4. log

x 20 � 2.5

5. logx 3.6 � 4 6. log

x 31.2 � 3.1

7. logx 2.34 � 20.75 8. log

x 9 � √

__

2

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Problem 4 More SolvingSolve each logarithmic equation by converting to an exponential equation and

applying the properties of exponents.

1. log 9,425 � x 2. log6 x � 3.41

3. ln x � 4 √__

3 4. logx 67.78 � 5.67

Be prepared to share your methods and solutions.

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Lesson 8.2 l Logarithms and Problem Solving Part I 313

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Problem 1 DecibelsA decibel is a unit of measure for the loudness of sound. The formula for the loudness

of a sound is given by

dB � 10 log I __

I0

where dB is the decibel level. The quantity I0 is the intensity of the threshold

sound, a sound that can barely be perceived. The intensity of other sounds,

I, are defined as the number of times more intense they are than

threshold sound.

1. The sound in a quiet library is about 1000 times as intense as the

threshold sound, I � 1000I0. What is the decibel level of a quiet library?


l Use logarithms to solve problems.

8.2 Decibels, pH, and the Richter ScaleLogarithms and Problem Solving Part I

Key Termsl decibel

l pH scale

l Richter scale

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2. The dial tone on a telephone has a decibel level of 80. A dial tone is how

many times more intense than the threshold sound?

3. Prolonged exposure to sounds above 85 decibels can cause hearing

damage or loss. A loud rock concert averages about 115 decibels.

A rock concert is how many times more intense than the threshold sound?

4. A sound of a motorcycle is 100,000,000,000 times more intense than a

threshold sound. What is the decibel level for a motorcycle?

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Problem 2 pHThe pH scale is a scale for measuring the acidity or alkalinity of a substance,

which is determined by the concentration of hydrogen ions. The formula for

pH is

pH � �log H�

where H� is the concentration of hydrogen ions. Solutions with a pH value of

less than 7 are acidic. Solutions with a pH value greater than 7 are alkaline or

basic. Solutions with a pH of 7 are neutral. For example, plain water has a

pH of 7.

1. The H� concentration in orange juice is 0.000199. What is the pH level of

the orange juice? Is orange juice acidic or basic?

2. The concentration of hydrogen ions in baking soda is 5.012 � 10�9.

What is the pH level of baking soda? Is baking soda acidic or basic?

3. Vinegar has a pH of 2.2. What is the concentration of hydrogen ions

in vinegar?

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4. Lime water has a pH of 12. What is the concentration of hydrogen ions

in lime water?

Problem 3 Richter ScaleThe Richter scale is a scale used to measure the intensity of earthquakes.

A seismograph is an instrument that measures the motion of the ground.

The formula for calculating the Richter scale is

M � log ( I __ I0

) where M is the magnitude of the earthquake on the Richter scale, usually

rounded to the nearest tenth. The quantity I0 represents the intensity of

a zero-level earthquake the same distance from the epicenter. The

seismographic reading for a zero-level earthquake is 0.001 millimeter at

a distance of 100 kilometers from the center of the earthquake. The intensity

of other earthquakes, I, are defined as the number of times more intense they

are than a zero-level earthquake.

1. The San Francisco earthquake of 1906 had a seismographic reading of

7943 millimeters registered 100 kilometers from the center. What was the

magnitude of the San Francisco earthquake of 1906 on the Richter scale?

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2. The San Francisco earthquake of October 17, 1989 struck the Bay Area

just before the third game of the World Series at Candlestick Park. It was

the worst earthquake since 1906. Its seismographic reading of

12,589 millimeters was registered 100 kilometers from the center.

What was the magnitude of the San Francisco earthquake of 1989 on

the Richter scale?

3. The great Alaska earthquake on March 27, 1940 had a magnitude of 9.2

on the Richter scale. What was its seismographic reading in millimeters

100 kilometers from the center?

4. Calculate the value of the seismographic reading for an earthquake of

magnitude 7 on the Richter scale.

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5. Calculate the value of the seismographic reading for an earthquake of

magnitude 8 on the Richter scale.

6. How much greater is the motion between a magnitude 8 earthquake and a

magnitude 7 earthquake? What would you predict as the amount of damage

caused by a magnitude 8 compared to a magnitude 7 earthquake?


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Lesson 8.3 l Logarithms and Problem Solving Part II 319

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8.3 Depreciation, Population Growth, and Radioactive DecayLogarithms and Problem Solving Part II


l Use logarithms to solve real-world problems.

Problem 1 DepreciationSome items, such as automobiles, are worth less over time. The age of an item

can be predicted using the formula

t � log ( V __

C ) _________

log(1 � r )

where t is the age of the item in years, V is the value of the item after t years,

C is the original value of the item, and r is the yearly rate of depreciation

expressed as a decimal.

1. A sports car was originally purchased for $42,750 and is currently valued

at $23,350. The average rate of depreciation for this car is 11.6% per year.

How old is the car to the nearest tenth of a year?

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2. A compact car was originally purchased for $15,450 and is currently

valued at $5250. The average rate of depreciation for this car is

25.5% per year. How old is this car to the nearest tenth of a year?

3. A 5-year-old car was originally purchased for $27,450. Its current value

is $12,250. What is this car’s annual rate of depreciation?

4. A 6-year-old car is currently valued at $10,250. The car depreciates at

the rate of 27.5% per year. What was the original price of this car?

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Problem 2 Population Growth 1. Some biologists study the population of a species in certain regions.

The formula for the population of a species is

n � k log(A)

where n represents the population of a species, A is the area of the

region in which the species lives, and k is a constant that is determined

by field studies. Based on population samples, a rainforest that is

1000 square miles has 2400 parrots.

a. What is the value of k?

b. Based on the current level of deforestation, it is estimated that in

10 years only about 200 square miles of the rainforest will remain.

How many parrots will live in the rainforest in 10 years?

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2. Desalination is a process for producing fresh water from salt water. The

amount of fresh water produced can be modeled using the formula

y � a � b In t

where t represents the time in hours, y represents the amount of fresh

water produced in t hours, a represents the amount of fresh water

produced in one hour, and b is the rate of production.

a. In one desalination plant, 12.78 cubic yards of fresh water can be

produced in one hour with a rate of production of 25.6. How much

fresh water can be produced after 5 hours?

b. How long would it take for the plant to produce 100 cubic yards of

fresh water?

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3. The amount of medicine left in a patient’s body can be predicted by

the formula

t � log ( C __

A ) _________

log(1 � r )

where t is the time in hours since the medicine was administered,

C is the current amount of the medicine left in the patient’s body in

milligrams, A is the original dose of the medicine in milligrams, and r is

the rate at which the medicine leaves the body.

a. A patient is given 10 milligrams of a medicine which leaves the body

at the rate of 20% per hour. How long will it take for 2 milligrams of

the medicine to remain in the patient’s body?

b. Six hours after administering a 20 milligram dose of medicine,

5 milligrams remain in the patient’s body. At what rate is the medicine

leaving the body?

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Problem 3 Radioactive DecayA radioactive isotope decays over time. The amount of a radioactive isotope

remaining can be modeled using the formula

A � A0 e�kt

where t represents the time in years, A represents the amount of the isotope

remaining in grams after t years, A0 represents the original amount of the

isotope in grams, and k is a decay constant.

The half-life, �, of a radioactive isotope is the time required for half of the

sample to decay. The relationship between k and � is determined by the

assumption that a sample of A0 grams will contain

1 __

2 A

0 grams after t years.

The half-lives of some common radioactive isotopes are shown in the table.

Isotope Half-life, �Decay

Constant, kDecay

Formula

Uranium (U-238) 4,510,000,000 years

Plutonium (Pu-239) 24,360 years

Carbon (C-14) 5730 years

1. Determine an equation for the decay constant in terms of the half-life by

substituting 1 __

2 A

0 for A and solving for k.

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2. Calculate the decay constant for each radioactive isotope in the table.

Enter each decay constant in the table.

3. Write a decay formula for each radioactive isotope by substituting

each decay constant into the general decay formula. Enter each decay

formula in the table.

4. Calculate the percentage of Plutonium 239 remaining after

100,000 years.

5. Calculate the amount of 100 grams of Carbon 14 remaining after

10,000 years.


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Lesson 8.4 l Loans and Investments 327

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8.4 Money, Money, Money!Loans and Investments

Problem 1 Car LoansOne of the first major purchases that a person makes is buying a car. Many

people purchase cars by taking out a loan and making monthly payments until

the loan is repaid with interest. The amount of the loan still owed is called the

principal, the amount paid each month is the payment, and the interest rate is

usually a yearly rate. Most car loans are for three to five years.

You want to purchase a $16,500 car with a $1000 down payment.

1. What is the amount of the loan that you would need to purchase

the car?

The car dealership offers three different financing options.

l A three-year loan at 3.6%

l A four-year loan at 4.6%

l A five-year loan at 5.6%

The loan that is best for you depends on several considerations including:

l How long do you want the loan to last?

l What monthly payment can you afford?

l How long are you planning on keeping the car?

The monthly payment for a car loan can be determined using the formula

p � iP

0 ____________ 1 � (1 � i )�n

where i is the interest rate per payment period, P0 is the original value of the

loan, p is the monthly payment, and n is the number of payments.

ObjectivesIn this lesson you will:

l Calculate the payment on a loan.

l Calculate the total paid on a loan.

l Calculate the number of payments.

l Calculate the time it takes for an investment to reach a particular amount.

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2. Calculate the monthly payment for each loan. Then calculate the total

amount paid.

a. A three-year loan at 3.6%

b. A four-year loan at 4.6%

c. A five-year loan at 5.6%

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3. Based on the information in Question 2, which loan would you choose?

Explain.

4. Your friend Herman heard that if you take a loan and pay more than the

minimum payment you can save a lot of money. The number of monthly

payments can be calculated using the formula

n � �log ( 1 �

iP0 ___ p ) _____________

log(1 � i )

where i is the interest rate per payment period, P0 is the original value of

the loan, p is the monthly payment, and n is the number of payments.

a. How many payments would you make on the four-year loan if you

paid $500 each month instead of the minimum payment?

b. What is the total amount paid?

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c. How many payments of $500 would you make? What would be the

amount of the last payment?

d. How much money would you save by paying $500 each month

instead of the minimum payment?

5. For each monthly payment amount on the five-year loan, calculate the

number of monthly payments, the amount of the last payment, and the

total amount paid.

a. $300

b. $310

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Problem 2 MortgagesBesides a car, the largest purchase that many people make is a house,

condominium, or apartment. The price of a home is usually much greater than

the price of a car so nearly everyone takes out a mortgage or home loan of 60%

to 95% of the selling price of the home. A standard mortgage usually requires a

down payment of 20% of the selling price and is for a time period from 20 to

30 years. The median national price of a house in the year 2000 was $139,000.

1. You want to purchase a house that is the median cost. What would

be the amount of the down payment? What would be the amount of

the loan?

2. You secure a 30-year mortgage loan at an annual percentage interest

rate (APR) of 5.378%.

a. What would be your monthly payment?

b. What is the total amount that you would pay against the loan?

c. How much interest would you pay over the length of the loan?

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3. You want your monthly payments on the 30-year loan to be $650.

a. How many payments will you need to make?

b. What is the total amount of the payments?

c. How much is the amount of the last payment?

d. How much interest would you save?

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Problem 3 InvestmentsYou just got your first job. Your uncle Robert tells you that it’s never too early

to start planning for your retirement. He says you should start putting $10 into

a savings account each month and in 40 years when you retire you will have

thousands of dollars.

You find a savings account that pays 3.2% monthly. The amount of money in

the account can be calculated using the formula

P � p

__ i [(1 � i )n � 1]

where P is the amount of money in the account, p is the amount that you

deposit each month, n is the number of months, and i is the interest per

payment period.

1. How much money would you have saved after 5 years?

2. How much money would you have saved after 40 years?

3. Was your uncle Robert correct? Would this be enough to retire? Explain.

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4. You deposit $10 each week instead of each month with the interest

compounded weekly. How much money would you have saved after

40 years?

You want to have $1,000,000 in 45 years when you retire. You want to know

how many monthly deposits of $200 in this account will it take reach this goal.

The formula for the number of months to reach an investment goal is

n � log ( 1 �

iF� ___ p ) ___________

log(1 � i )

where F� is the future value or investment goal, p is the amount that you

deposit each month, n is the number of months, and i is the interest per

payment period.

5. How many payments of $200 a month would it take to reach your goal

of $1,000,000? How many years is this?

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6. How many payments of $500 a month would it take to reach your goal

of $1,000,000? How many years is this?

7. What monthly payment would allow you to make your goal of $1,000,000

in 45 years?


8 logarithmic functions exponential and - tench's...

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