functions - edward c. reed high...

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Name. .Class- Date-

Functionsiofeg Seepeo3

Essential question: What are the characteristics of logarithmic functions?

Recall that if/(x) is a one-to-one function, then the graphs of/(x) and itsinverse,/'~\x}, are reflections of each other about the line j/ = x. The domainof/(X) is the range off 1(x), and the range of/(x) is the domain off~1(x).

E X P L O R E \g the Inverse of ao Exponential Function

The graph of /(x) = 2X is shown. Graph r"~1(x) by following these steps.

A Complete the table by writing the image of each point on the graphofj[x) after a reflection across the line y = x.

Point on thegraph of f(x)

(-2,0.25) —

(-1,0.5) —

(0, 1) —

(1, 2) —

(2, 4) —

Point on thegraph's image

-*- (0.25, -2)

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Video Tutor

B Plot and label the image of each point on thecoordinate plane.

C Use the images of the points to sketch the graph of/~1(x).

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REFLECT

| 1a. What are the domain and range of/ *(x)?

! 1b. Does the graph of/ (x) have any asymptotes? Explain how you know.

1c:. How do the values of/ (x) change as x increases without bound?

Chapter 4 207 Lesson 3

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E N G A G E Defining Logarithmic Fractions

A logarithm is the exponent to which a base must be raised in order to obtain a givenvalue. For example, 23 = 8, so the logarithm base 2 of 8 is 3, and you write log, 8 = 3.

Definition of Loqariti

For positive numbers y and b (b ̂ 1), the logarithm of y with base b iswritten logfoy and is defined as follows:

logfo y = x if and only ifbx = y.\^_^. _^ __-''

This definition means that every statement about exponents can be converted into anequivalent statement about logarithms, and vice versa. Note that you read log^x as"the logarithm base b of x" or "log base b ofx."

A logarithmic function with base b is the inverse of the exponential function with base b.For instance, the inverse of/(x) = 2X is/-1(jc) = Iog2 x, the graph of which yousketched in the Explore.

The table describes two special logarithms.

Special Logarithms

Name

Common logarithm

Natural logarithm

Base

10

e

Notation

Write log x instead of Iog10 x.

Write In x instead of logex.

The number e isthe value that thex

expression (1 + -£)approaches as x increaseswithout bound, e isan irrational number,approximately 2.72.

2a. Explain, in terms of a logarithmic function, how to write 72 = 49 as an equivalentstatement involving a logarithm.

2b. Explain, in terms of an exponential function, how to write log 1000 = 3 as anequivalent statement involving an exponent.

2c. The input of/(x) = 2X is an exponent and the output is a power of 2. Describe theinput and output off~1(x) = Iog2 x. Give a specific example.

2d. Find In ~ by letting In \ x and writing this statement in an equivalent form thatinvolves an exponent. Explain your reasoning from that point on.


E X A M P L E \g Logarithmic Factions

Find each value of f(x) = Iog2x.

A /(16)

Write the function's input as a power of 2. The function's output is the exponent.

-

16 = 2 , so/(16)=

B /(64)

64 = 2 ,so/(64) =

C

,so/{i)=.

1 = 2 ,so/g)=

/(I)

1 = 2 ,so/l)=

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3a. Is it possible to evaluate/(O)? Why or why not?

3b. foif(x) - Iog2x, between which two integers does/(40) lie? Explain.

3c. Estimate g(95) for g(x) = \ogx, without using a calculator. Explain.

3d. Without using a calculator, explain how you know that In 20 > log 20.


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E X A M P L E \g a Logarithmic

Graph f(x) = \og,x.

A What is the range of/(x)? (Think: what powers can you raise | to?) .What is the domain of/(jc)? (Think: what values can be obtained

by raising | to a power?)

B Complete the table of values. C Plot the points. Connect them with a smooth curve.

f(x)

4a. How is the graph off(x) — logj x related to the graph oif(x) — Iog2x from the Explore?2

Why does this make sense?

4b. What point do the graphs of/(x) = logj x and/(x) = Iog2 x have in common? Why?

4c. Describe the end behavior off(x) = logj x.

4d. How is the graph of/(x) = logj x related to the graph of/(x) = f ̂ J ?


PRACTICE

1. The graph of/(x) = 3* is shown.

a. Use the labeled points to help you draw the graphoff~i(x). Label the corresponding points onf~l(x).

b. Write the inverse function, f~\x}, using logarithmicnotation.

c. State the domain and range of/ l(x).

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Find each value of f(x) = Iog4x.

2- /(16) 3.

Find each value of f(x) = log^.4

8. /(16)

Find each value of f(x) = logx.

5. /(10,000) 6. /(O.I)

9. /(I)

Evaluate each expression.

11. Iog864 12. Iog21024

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13. log 1,000,000

14. For/(x) = log x, between what two integers does/(6) lie? Explain.

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15. Explain how you can estimate the value of In 10 without using a calculator.

16. What is the value of log, b for b > 0 and b ± 1? Explain.


Graph each logarithmic function.

—-—2

L4J^-4-i~4—a»I J 1 ' ' ' ;i I

19. The graph of what logarithmic function is shown? Explainyour reasoning.

20. Name some values you would choose for x if you wereplotting points to sketch the graph of/(x) = log x withoutusing a calculator. Explain.

21. The graph of/(x) = logfex passes through the points (1, 0) and (36, 2). What is thevalue of bl Explain.

22. When you fold a sheet of paper in half x times, the function/(x) = 2* gives thenumber of sections that are created by the folds.

a. Describe the input and output of the function/"1^) = Iog2xin the problem context.

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b. Use/ (x) to find the number of folds needed to create 64 sections.

e. Assume that a sheet of paper has an area of 1. Write an exponential function g(x)that gives the area of a section of the paper after being folded in half xtimes.

d. Write the rule for g :(x) and describe the function's input and output.


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Name- Class. Date

j Additional Practice/Write each exponential equation in logarithmic form.

. 37 = 2187 2. 122 = 144 3. 53 = 125

Write each logarithmic equation in exponential form.

4. logio 100,000 = 5 5. Iog41024 = 5 6. Iog9729 =

Evaluate by using mental math.

7. log 1,000,000 8. log 10 9. log 1

10. Iog416 11. Iog8 1 12. Iog5 625

Use the given x-values to graph each function. Then graph itsinverse. Describe the domain and range of the inverse function.

13. f(x) = 2*;x = -2, -1,0, 1 ,2 , 3,4 14. f(x) = \x = -3,-2I-1,0, 1,2, 3

i"

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2 4 9

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3 1P 1? 14 16

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: i1,10

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2 4 6 $ 10

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Solve.

15. The hydrogen ion concentration in moles per liter for a certainbrand of tomato-vegetable juice is 0.000316.

a. Write a logarithmic equation for the pH of the juice.

b. What is the pH of the juice?


Problem Solving/

The acidity of rainwater varies from location to location. Acidity ismeasured in pH and is given by the function pH = -log[H+], where[H+] represents the hydrogen ion concentration in moles per liter. Thetable gives the [H*] of rainwater in different locations.

1. Find the acidity of rainwater in eastern Ohio.

a. Substitute the hydrogen ion concentrationfor rainwater in eastern Ohio in the functionfor pH.

b. Evaluate the function. What is the acidityof rainwater in eastern Ohio to the nearesttenth of a unit?

Hydrogen Ion Concentrationof Rainwater

Location

Central California

Eastern Texas

Eastern Ohio

[HI(moles per liter)

0.0000032

0.0000192

0.0000629

2. Find how the acidity of rainwater in centralCalifornia compares to the acidity of rainwater in eastern Ohio.

a. Write a function for the acidity of rainwater incentral California.

b. Evaluate the function. What is the acidity ofrainwater in central California to the nearesttenth of a unit?

c. Compare the pH of rainwater in the twolocations. Is the pH of rainwater in easternOhio greater than or less than that in centralCalifornia? By how much?

Choose the letter for the best answer.

3. What is the pH of rainwater in easternTexas?

A pH = 3.7 C pH = 4.4

B pH=4.0 D pH = 4.7

5. What is the pH of a sample of irrigationwater with a hydrogen ion concentrationof 8.3 x 10~7 moles per liter?

A pH = 6.1 C pH = 6.3

B pH = 6.2 D pH = 6.4

4. Nick makes his own vegetable juice. Ithas a hydrogen ion concentration of5.9 x 10~6 moles per liter. What is the pHof his vegetable juice?

F pH = 4.9 H pH = 5.1

G pH = 5.0 J pH-5.2

6. What is the pH of a shampoo sample witha hydrogen ion concentration of1.7x 10~8 moles per liter?

F pH = 7.4 H pH = 7.8

G pH = 7.6 J pH = 8.0

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