11
Introduction So far in this chapter, you have studied the definitions, graphs, and properties of exponential and logarithmic functions. In this section, you will study procedures for solving equations involving these exponential and logarithmic functions. There are two basic strategies for solving exponential or logarithmic equations. The first is based on the One-to-One Properties and was used to solve simple exponential and logarithmic equations in Sections 3.1 and 3.2. The second is based on the Inverse Properties. For and the following properties are true for all and for which and are defined. One-to-One Properties if and only if if and only if Inverse Properties Solving Simple Equations Original Rewritten Equation Equation Solution Property a. One-to-One b. One-to-One c. One-to-One d. Inverse e. Inverse f. Inverse Now try Exercise 13. The strategies used in Example 1 are summarized as follows. x 10 1 1 10 10 log x 10 1 log x 1 x e 3 e ln x e 3 ln x 3 x ln 7 ln e x ln 7 e x 7 x 2 3 x 3 2 1 3 x 9 x 3 ln x ln 3 ln x ln 3 0 x 5 2 x 2 5 2 x 32 log a a x x a log a x x x y. log a x log a y x y. a x a y log a y log a x y x a 1, a > 0 246 Chapter 3 Exponential and Logarithmic Functions What you should learn Solve simple exponential and logarithmic equations. Solve more complicated exponential equations. Solve more complicated logarithmic equations. Use exponential and logarith- mic equations to model and solve real-life problems. Why you should learn it Exponential and logarithmic equations are used to model and solve life science applica- tions. For instance, in Exercise 112, on page 255, a logarithmic function is used to model the number of trees per acre given the average diameter of the trees. Exponential and Logarithmic Equations © James Marshall/Corbis 3.4 Strategies for Solving Exponential and Logarithmic Equations 1. Rewrite the original equation in a form that allows the use of the One-to-One Properties of exponential or logarithmic functions. 2. Rewrite an exponential equation in logarithmic form and apply the Inverse Property of logarithmic functions. 3. Rewrite a logarithmic equation in exponential form and apply the Inverse Property of exponential functions. Example 1 333202_0304.qxd 12/7/05 10:31 AM Page 246

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Page 1: 3.4 Exponential and Logarithmic Equations and logarithmic functions. ... simple exponential and logarithmic equations in Sections 3.1 and 3.2 ... 246 Chapter 3 Exponential and Logarithmic

IntroductionSo far in this chapter, you have studied the definitions, graphs, and properties ofexponential and logarithmic functions. In this section, you will study proceduresfor solving equations involving these exponential and logarithmic functions.

There are two basic strategies for solving exponential or logarithmicequations. The first is based on the One-to-One Properties and was used to solvesimple exponential and logarithmic equations in Sections 3.1 and 3.2. Thesecond is based on the Inverse Properties. For and the followingproperties are true for all and for which and are defined.

One-to-One Properties

if and only if

if and only if

Inverse Properties

Solving Simple Equations

Original RewrittenEquation Equation Solution Property

a. One-to-One

b. One-to-One

c. One-to-One

d. Inverse

e. Inverse

f. Inverse

Now try Exercise 13.

The strategies used in Example 1 are summarized as follows.

x � 10�1 �11010 log x � 10�1log x � �1

x � e�3e ln x � e�3ln x � �3

x � ln 7ln ex � ln 7ex � 7

x � �23�x � 32�13�x

� 9

x � 3ln x � ln 3ln x � ln 3 � 0

x � 52x � 252x � 32

loga ax � x

aloga x � x

x � y.loga x � loga y

x � y.ax � ay

loga yloga xyxa � 1,a > 0

246 Chapter 3 Exponential and Logarithmic Functions

What you should learn• Solve simple exponential

and logarithmic equations.

• Solve more complicated exponential equations.

• Solve more complicated logarithmic equations.

• Use exponential and logarith-mic equations to model andsolve real-life problems.

Why you should learn itExponential and logarithmicequations are used to modeland solve life science applica-tions. For instance, in Exercise112, on page 255, a logarithmicfunction is used to model thenumber of trees per acre giventhe average diameter of thetrees.

Exponential and Logarithmic Equations

© James Marshall/Corbis

3.4

Strategies for Solving Exponential and LogarithmicEquations

1. Rewrite the original equation in a form that allows the use of the One-to-One Properties of exponential or logarithmic functions.

2. Rewrite an exponential equation in logarithmic form and apply theInverse Property of logarithmic functions.

3. Rewrite a logarithmic equation in exponential form and apply the InverseProperty of exponential functions.

Example 1

333202_0304.qxd 12/7/05 10:31 AM Page 246

Page 2: 3.4 Exponential and Logarithmic Equations and logarithmic functions. ... simple exponential and logarithmic equations in Sections 3.1 and 3.2 ... 246 Chapter 3 Exponential and Logarithmic

Solving Exponential Equations

Solving Exponential Equations

Solve each equation and approximate the result to three decimal places if necessary.

a. b.

Solutiona. Write original equation.

One-to-One Property

Write in general form.

Factor.

Set 1st factor equal to 0.

Set 2nd factor equal to 0.

The solutions are and Check these in the original equation.

b. Write original equation.

Divide each side by 3.

Take log (base 2) of each side.

Inverse Property

Change-of-base formula

The solution is Check this in the original equation.

Now try Exercise 25.

In Example 2(b), the exact solution is and the approximatesolution is An exact answer is preferred when the solution is anintermediate step in a larger problem. For a final answer, an approximate solutionis easier to comprehend.

Solving an Exponential Equation

Solve and approximate the result to three decimal places.

SolutionWrite original equation.

Subtract 5 from each side.

Take natural log of each side.

Inverse Property

The solution is ln 55 . Check this in the original equation.

Now try Exercise 51.

� 4.007x �

x � ln 55 � 4.007

ln ex � ln 55

ex � 55

ex � 5 � 60

ex � 5 � 60

x � 3.807.x � log2 14

x � log2 14 � 3.807.

x �ln 14ln 2

� 3.807

x � log2 14

log2 2x � log2 14

2 x � 14

3�2 x� � 42

x � 4.x � �1

�x � 4� � 0 ⇒ x � 4

�x � 1� � 0 ⇒ x � �1

�x � 1��x � 4� � 0

x2 � 3x � 4 � 0

�x2 � �3x � 4

e�x2� e�3x�4

3�2 x� � 42e�x2� e�3x�4

Section 3.4 Exponential and Logarithmic Equations 247

Remember that the natural loga-rithmic function has a base of e.

Example 2

Example 3

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Page 3: 3.4 Exponential and Logarithmic Equations and logarithmic functions. ... simple exponential and logarithmic equations in Sections 3.1 and 3.2 ... 246 Chapter 3 Exponential and Logarithmic

Solving an Exponential Equation

Solve and approximate the result to three decimal places.

SolutionWrite original equation.

Add 4 to each side.

Divide each side by 2.

Take log (base 3) of each side.

Inverse Property

Add 5 to each side.

Divide each side by 2.

Use a calculator.

The solution is Check this in the original equation.

Now try Exercise 53.

When an equation involves two or more exponential expressions, you canstill use a procedure similar to that demonstrated in Examples 2, 3, and 4.However, the algebra is a bit more complicated.

t �52 �

12 log3 7.5 � 3.417.

t � 3.417

t �52

�12

log3 7.5

2t � 5 � log3 7.5

2t � 5 � log3 152

log3 32t�5 � log3

152

32t�5 �152

2�32t�5� � 15

2�32t�5� � 4 � 11

2�32t�5� � 4 � 11

248 Chapter 3 Exponential and Logarithmic Functions

Example 4Additional Example

Solve

Solution

Note: Using the change-of-baseformula or the definition of a logarith-mic function, you could write thissolution as x � log2 10.

x �ln 10ln 2

� 3.322

x ln 2 � ln 10

ln 2x � ln 10

2x � 10

2x � 10.

To ensure that students first solve forthe unknown variable algebraically andthen use their calculators, you canrequire both exact algebraic solutionsand approximate numerical answers.

Remember that to evaluate alogarithm such as youneed to use the change-of-baseformula.

log3 7.5 �ln 7.5ln 3

� 1.834

log3 7.5,

Solving an Exponential Equation of Quadratic Type

Solve e2x � 3ex � 2 � 0.

Example 5

Algebraic SolutionWrite original equation.

Write in quadratic form.

Factor.

Set 1st factor equal to 0.

Solution

Set 2nd factor equal to 0.

Solution

The solutions are and Checkthese in the original equation.

Now try Exercise 67.

x � 0.x � ln 2 � 0.693

x � 0

ex � 1 � 0

x � ln 2

ex � 2 � 0

�ex � 2��ex � 1� � 0

�ex�2 � 3ex � 2 � 0

e2x � 3ex � 2 � 0

Graphical SolutionUse a graphing utility to graph Use thezero or root feature or the zoom and trace features of thegraphing utility to approximate the values of for which

In Figure 3.25, you can see that the zeros occur atand at So, the solutions are and

FIGURE 3.25

3

−1

3

3y = e2x − 3ex + 2

x � 0.693.x � 0x � 0.693.x � 0

y � 0.x

y � e2x � 3ex � 2.

333202_0304.qxd 12/7/05 10:31 AM Page 248

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Solving Logarithmic EquationsTo solve a logarithmic equation, you can write it in exponential form.

Logarithmic form

Exponentiate each side.

Exponential form

This procedure is called exponentiating each side of an equation.

Solving Logarithmic Equations

a. Original equation

Exponentiate each side.

Inverse Property

b. Original equation

One-to-One Property

Add and 1 to each side.

Divide each side by 4.

c. Original equation

Quotient Property of Logarithms

One-to-One Property

Cross multiply.

Isolate

Divide each side by

Now try Exercise 77.

Solving a Logarithmic Equation

Solve and approximate the result to three decimal places.

SolutionWrite original equation.

Subtract 5 from each side.

Divide each side by 2.

Exponentiate each side.

Inverse Property

Use a calculator.

Now try Exercise 85.

x � 0.607

x � e�1�2

eln x � e�1�2

ln x � �1

2

2 ln x � �1

5 � 2 ln x � 4

5 � 2 ln x � 4

�7. x � 2

x. �7x � �14

3x � 14 � 10x

3x � 14

5� 2x

log6�3x � 145 � � log6 2x

log6�3x � 14� � log6 5 � log6 2x

x � 2

�x 4x � 8

5x � 1 � x � 7

log3�5x � 1� � log3�x � 7�

x � e2

e ln x � e2

ln x � 2

x � e3

eln x � e3

ln x � 3

Section 3.4 Exponential and Logarithmic Equations 249

Example 6

Example 7

Remember to check your solu-tions in the original equationwhen solving equations to verifythat the answer is correct and tomake sure that the answer lies in the domain of the originalequation.

Activities

1. Solve for

Answer:

2. Solve for

Answer: is not in thedomain.�

x � 1 �x � �6log�x � 4� � log�x � 1� � 1.

x:

x �ln 3ln 7

� 0.5646

7x � 3.x:

333202_0304.qxd 12/7/05 10:31 AM Page 249

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Solving a Logarithmic Equation

Solve 2

SolutionWrite original equation.

Divide each side by 2.

Exponentiate each side (base 5).

Inverse Property

Divide each side by 3.

The solution is Check this in the original equation.

Now try Exercise 87.

Because the domain of a logarithmic function generally does not include allreal numbers, you should be sure to check for extraneous solutions of logarithmicequations.

x �253 .

x �253

3x � 25

5 log5 3x � 52

log5 3x � 2

2 log5 3x � 4

log5 3x � 4.

250 Chapter 3 Exponential and Logarithmic Functions

Example 8

Notice in Example 9 that thelogarithmic part of the equationis condensed into a single loga-rithm before exponentiatingeach side of the equation.

In Example 9, the domain of is and the domain of isso the domain of the original equation is Because the domain is all

real numbers greater than 1, the solution is extraneous. The graph inFigure 3.26 verifies this concept.

x � �4x > 1.x > 1,

log�x � 1�x > 0log 5x

Checking for Extraneous Solutions

Solve log 5x � log�x � 1� � 2.

Example 9

Algebraic SolutionWrite original equation.

Product Property of Logarithms

Exponentiate each side (base 10).

Inverse Property

Write in general form.

Factor.

Set 1st factor equal to 0.

Solution

Set 2nd factor equal to 0.

Solution

The solutions appear to be and However, whenyou check these in the original equation, you can see that is the only solution.

Now try Exercise 99.

x � 5x � �4.x � 5

x � �4

x � 4 � 0

x � 5

x � 5 � 0

�x � 5��x � 4� � 0

x2 � x � 20 � 0

5x2 � 5x � 100

10 log�5x2�5x� � 102

log�5x�x � 1�� � 2

log 5x � log�x � 1� � 2

Graphical SolutionUse a graphing utility to graph

and in the sameviewing window. From the graph shown in Figure3.26, it appears that the graphs intersect at one point.Use the intersect feature or the zoom and tracefeatures to determine that the graphs intersect atapproximately So, the solution is Verify that 5 is an exact solution algebraically.

FIGURE 3.26

0

−1

9

y1 = log 5x + log(x − 1)

y2 = 2

5

x � 5.�5, 2�.

y2 � 2y1 � log 5x � log�x � 1�

333202_0304.qxd 12/7/05 10:31 AM Page 250

Page 6: 3.4 Exponential and Logarithmic Equations and logarithmic functions. ... simple exponential and logarithmic equations in Sections 3.1 and 3.2 ... 246 Chapter 3 Exponential and Logarithmic

Applications

Doubling an Investment

You have deposited $500 in an account that pays 6.75% interest, compoundedcontinuously. How long will it take your money to double?

SolutionUsing the formula for continuous compounding, you can find that the balance inthe account is

To find the time required for the balance to double, let and solve theresulting equation for

Let

Divide each side by 500.

Take natural log of each side.

Inverse Property

Divide each side by 0.0675.

Use a calculator.

The balance in the account will double after approximately 10.27 years. Thisresult is demonstrated graphically in Figure 3.27.

FIGURE 3.27

Now try Exercise 107.

In Example 10, an approximate answer of 10.27 years is given. Within thecontext of the problem, the exact solution, years, does not makesense as an answer.

�ln 2��0.0675

t

Time (in years)

(0, 500)

(10.27, 1000)

Acc

ount

bal

ance

(in

dol

lars

)

100

300

500

700

900

1100

108642

A

THE UNITED STATES OF AMERICA

THE UNITED STATES OF AMERICA

THE UNITED STATES OF AMERICA

WASHIN

GTON

WASHINGTON, D.C.

AC 4

C 31

1

1

1

1SERIES

1993

Doubling an Investment

A = 500e0.0675t

t � 10.27

t �ln 2

0.0675

0.0675t � ln 2

ln e0.0675t � ln 2

e0.0675t � 2

A � 1000. 500e0.0675t � 1000

t.A � 1000

A � 500e0.0675t.

A � Pert

Section 3.4 Exponential and Logarithmic Equations 251

Example 10

Activity

Determine the amount of time it wouldtake $1000 to double in an account thatpays 6.75% interest, compoundedcontinuously. How does this compare toExample 10?

Answer: 10.27 years; it takes the sameamount of time.

The effective yield of a savingsplan is the percent increase inthe balance after 1 year. Findthe effective yield for eachsavings plan when $1000 isdeposited in a savings account.

a. 7% annual interest rate,compounded annually

b. 7% annual interest rate,compounded continuously

c. 7% annual interest rate,compounded quarterly

d. 7.25% annual interest rate,compounded quarterly

Which savings plan has thegreatest effective yield? Whichsavings plan will have thehighest balance after 5 years?

Exploration

333202_0304.qxd 12/7/05 10:31 AM Page 251

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W RITING ABOUT MATHEMATICS

Endangered Animals

The number of endangered animal species in the United States from 1990 to2002 can be modeled by

where represents the year, with corresponding to 1990 (see Figure 3.28).During which year did the number of endangered animal species reach 357?(Source: U.S. Fish and Wildlife Service)

SolutionWrite original equation.

Substitute 357 for

Add 119 to each side.

Divide each side by 164.

Exponentiate each side.

Inverse Property

Use a calculator.

The solution is Because represents 1990, it follows that thenumber of endangered animals reached 357 in 1998.

Now try Exercise 113.

t � 10t � 18.

t � 18

t � e476�164

eln t � e476�164

ln t �476164

164 ln t � 476

y. �119 � 164 ln t � 357

�119 � 164 ln t � y

t � 10t

10 ≤ t ≤ 22y � �119 � 164 ln t,

y

252 Chapter 3 Exponential and Logarithmic Functions

Example 11

10 12 14 16 18 20 22

200

250

300

350

400

450

t

y

Endangered Animal Species

Num

ber

of s

peci

es

Year (10 ↔ 1990)

FIGURE 3.28

Comparing Mathematical Models The table shows the U.S.Postal Service rates y for sending an express mail packagefor selected years from 1985 through 2002, where represents 1985. (Source: U.S. Postal Service)

a. Create a scatter plot of the data. Find a linear model forthe data, and add its graph to your scatter plot. Accordingto this model, when will the rate for sending an expressmail package reach $19.00?

b. Create a new table showing values for ln x and ln y andcreate a scatter plot of these transformed data. Use themethod illustrated in Example 7 in Section 3.3 to find amodel for the transformed data, and add its graph toyour scatter plot. According to this model, when will therate for sending an express mail package reach $19.00?

c. Solve the model in part (b) for y, and add its graph toyour scatter plot in part (a). Which model better fits theoriginal data? Which model will better predict futurerates? Explain.

x � 5

Year, x Rate, y

5 10.75

8 12.00

11 13.95

15 15.00

19 15.75

21 16.00

22 17.85

333202_0304.qxd 12/7/05 10:31 AM Page 252

Page 8: 3.4 Exponential and Logarithmic Equations and logarithmic functions. ... simple exponential and logarithmic equations in Sections 3.1 and 3.2 ... 246 Chapter 3 Exponential and Logarithmic

Section 3.4 Exponential and Logarithmic Equations 253

Exercises 3.4

In Exercises 1–8, determine whether each -value is asolution (or an approximate solution) of the equation.

1. 2.

(a) (a)

(b) (b)

3.

(a)

(b)

(c)

4.

(a)

(b)

(c)

5.

(a)

(b)

(c)

6.

(a)

(b)

(c)

7.

(a)

(b)

(c)

8.

(a)

(b)

(c)

In Exercises 9–20, solve for

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

In Exercises 21–24, approximate the point of intersectionof the graphs of and Then solve the equation

algebraically to verify your approximation.

21. 22.

23. 24.

x

−48 12

4

8

12

fg

y

xf

g

y

4 8 12

4

g�x� � 0g�x� � 2

f �x� � ln�x � 4�f �x� � log3 x

x−8 −4

−44 8

4

8

12

f

g

y

x−8 −4

−44 8

4

12

f

g

y

g�x� � 9g�x� � 8

f �x� � 27xf �x� � 2x

f x � gxg.f

log5 x � �3log4 x � 3

ln x � �7ln x � �1

ex � 4ex � 2

ln x � ln 5 � 0ln x � ln 2 � 0

�14�x

� 64�12�x

� 32

3x � 2434x � 16

x.

x � 1 � ln 3.8

x � 45.701

x � 1 � e3.8

ln�x � 1� � 3.8

x � 163.650

x �12��3 � e5.8�

x �12��3 � ln 5.8�

ln�2x � 3� � 5.8

x � 102 � 3

x � 17

x � 1021

log2�x � 3� � 10

x �643

x � �4

x � 21.333

log4�3x� � 3

x � �0.0416

x �ln 6

5 ln 2

x �15��2 � ln 6�

2e5x�2 � 12

x � 1.219

x � �2 � ln 25

x � �2 � e25

3ex�2 � 75

x � 2x � 2

x � �1x � 5

23x�1 � 3242x�7 � 64

x

VOCABULARY CHECK: Fill in the blanks.

1. To ________ an equation in means to find all values of for which the equation is true.

2. To solve exponential and logarithmic equations, you can use the following One-to-One and Inverse Properties.

(a) if and only if ________.

(b) if and only if ________.

(c) ________

(d) ________

3. An ________ solution does not satisfy the original equation.

PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com.

loga ax �

aloga x �

loga x � loga y

ax � ay

xx

333202_0304.qxd 12/7/05 10:31 AM Page 253

Page 9: 3.4 Exponential and Logarithmic Equations and logarithmic functions. ... simple exponential and logarithmic equations in Sections 3.1 and 3.2 ... 246 Chapter 3 Exponential and Logarithmic

In Exercises 25–66, solve the exponential equationalgebraically. Approximate the result to three decimalplaces.

25. 26.

27. 28.

29. 30.

31. 32.

33. 34.

35. 36.

37. 38.

39. 40.

41. 42.

43. 44.

45. 46.

47. 48.

49. 50.

51. 52.

53. 54.

55. 56.

57. 58.

59. 60.

61. 62.

63. 64.

65. 66.

In Exercises 67–74, use a graphing utility to graph andsolve the equation. Approximate the result to threedecimal places. Verify your result algebraically.

67. 68.

69. 70.

71. 72.

73. 74.

In Exercises 75–102, solve the logarithmic equation alge-braically. Approximate the result to three decimal places.

75. 76.

77. 78.

79. 80.

81. 82.

83. 84.

85. 86.

87. 88.

89. 90.

91.

92.

93.

94.

95.

96.

97.

98.

99.

100.

101.

102.

In Exercises 103–106, use a graphing utility to graph andsolve the equation. Approximate the result to three decimal places. Verify your result algebraically.

103. 104.

105. 106.

Compound Interest In Exercises 107 and 108, $2500 isinvested in an account at interest rate compoundedcontinuously. Find the time required for the amount to (a) double and (b) triple.

107. 108.

109. Demand The demand equation for a microwave oven isgiven by

Find the demand for a price of (a) and (b)

110. Demand The demand equation for a hand-held elec-tronic organizer is

Find the demand for a price of (a) and (b)

111. Forest Yield The yield (in millions of cubic feet peracre) for a forest at age years is given by

(a) Use a graphing utility to graph the function.

(b) Determine the horizontal asymptote of the function.Interpret its meaning in the context of the problem.

(c) Find the time necessary to obtain a yield of 1.3 millioncubic feet.

V � 6.7e�48.1�t.

tV

p � $400.p � $600x

p � 5000�1 �4

4 � e�0.002x�.

p � $300.p � $350x

p � 500 � 0.5�e0.004x�.

r � 0.12r � 0.085

r,

10 � 4 ln�x � 2� � 03 � ln x � 0

500 � 1500e�x�27 � 2x

log 4x � log�12 � �x � � 2

log 8x � log�1 � �x � � 2

log3 x � log3�x � 8� � 2

log4 x � log4�x � 1� �12

log2 x � log2�x � 2� � log2�x � 6�log�x � 4� � log x � log�x � 2�log�x � 6� � log�2x � 1�log2�2x � 3� � log2�x � 4�ln�x � 1� � ln�x � 2� � ln x

ln�x � 5� � ln�x � 1� � ln�x � 1�ln x � ln�x � 3� � 1

ln x � ln�x � 2� � 1

ln x � ln�x � 1� � 1ln x � ln�x � 1� � 2

5 log10�x � 2� � 116 log3�0.5x� � 11

2 � 6 ln x � 107 � 3 ln x � 5

ln�x � 8 � 5ln�x � 2 � 1

2 ln x � 73ln 5x � 10

log 3z � 2log x � 6

ln 4x � 1ln 2x � 2.4

ln x � 2ln x � �3

e2.724x � 29e 0.125t � 8 � 0

�e1.8x � 7 � 0e0.09t � 3

8e�2x�3 � 113e3x�2 � 962

�4e�x�1 � 15 � 06e1�x � 25

�16 �0.878

26 �3t

� 30�1 �0.10

12 �12t

� 2

�4 �2.471

40 �9t

� 21�1 �0.065

365 �365t

� 4

119e6x � 14

� 73000

2 � e2x� 2

400

1 � e�x� 350

500100 � ex�2 � 20

e2x � 9ex � 36 � 0e2x � 3ex � 4 � 0

e2x � 5ex � 6 � 0e2x � 4ex � 5 � 0

8�46�2x� � 13 � 416�23x�1� � 7 � 9

�14 � 3ex � 117 � 2ex � 5

1000e�4x � 75500e�x � 300

e2x � 50e3x � 12

8�36�x� � 403�5x�1� � 21

5�10 x�6� � 78�103x� � 12

8�2�x � 43123�x � 565

2x�3 � 323x�1 � 27

4�3t � 0.105�t�2 � 0.20

65x � 300032x � 80

6x � 10 � 47ex � 9 � 19

4ex � 912ex � 10

2�5x� � 324�3x� � 20

e�x2� ex2�2xex2�3 � ex�2

e2x � ex2�8ex � ex2�2

254 Chapter 3 Exponential and Logarithmic Functions

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Section 3.4 Exponential and Logarithmic Equations 255

112. Trees per Acre The number of trees of a givenspecies per acre is approximated by the model

where is the averagediameter of the trees (in inches) 3 feet above the ground.Use the model to approximate the average diameter of thetrees in a test plot when

113. Medicine The number of hospitals in the UnitedStates from 1995 to 2002 can be modeled by

where represents the year, with corresponding to1995. During which year did the number of hospitalsreach 5800? (Source: Health Forum)

114. Sports The number of daily fee golf facilities in theUnited States from 1995 to 2003 can be modeled by

where representsthe year, with corresponding to 1995. During whichyear did the number of daily fee golf facilities reach9000? (Source: National Golf Foundation)

115. Average Heights The percent of American malesbetween the ages of 18 and 24 who are no more than

inches tall is modeled by

and the percent of American females between the ages of18 and 24 who are no more than inches tall is modeledby

(Source: U.S. National Center for Health Statistics)

(a) Use the graph to determine any horizontal asymptotesof the graphs of the functions. Interpret the meaningin the context of the problem.

(b) What is the average height of each sex?

116. Learning Curve In a group project in learning theory, amathematical model for the proportion of correctresponses after trials was found to be

(a) Use a graphing utility to graph the function.

(b) Use the graph to determine any horizontal asymptotesof the graph of the function. Interpret the meaning ofthe upper asymptote in the context of this problem.

(c) After how many trials will 60% of the responses becorrect?

P �0.83

1 � e�0.2n.

nP

Height (in inches)

Perc

ent o

fpo

pula

tion

x

f(x)

m(x)20

40

60

80

100

55 60 65 70 75

f �x� �100

1 � e�0.66607�x�64.51�.

xf

m�x� �100

1 � e�0.6114�x�69.71�

x

m

t � 5t5 ≤ t ≤ 13y � 4381 � 1883.6 ln t,

y

t � 5t

y � 7312 � 630.0 ln t, 5 ≤ t ≤ 12

y

N � 21.

x5 ≤ x ≤ 40N � 68�10�0.04x�,

N

x 0.2 0.4 0.6 0.8 1.0

y

117. Automobiles Automobiles are designed with crum-ple zones that help protect their occupants in crashes.The crumple zones allow the occupants to move shortdistances when the automobiles come to abrupt stops.The greater the distance moved, the fewer g’s the crashvictims experience. (One g is equal to the accelerationdue to gravity. For very short periods of time, humanshave withstood as much as 40 g’s.) In crash tests withvehicles moving at 90 kilometers per hour, analystsmeasured the numbers of g’s experienced duringdeceleration by crash dummies that were permitted tomove meters during impact. The data are shown inthe table.

A model for the data is given by

where is the number of g’s.

(a) Complete the table using the model.

(b) Use a graphing utility to graph the data points andthe model in the same viewing window. How dothey compare?

(c) Use the model to estimate the distance traveledduring impact if the passenger deceleration mustnot exceed 30 g’s.

(d) Do you think it is practical to lower the number ofg’s experienced during impact to fewer than 23?Explain your reasoning.

y

y � �3.00 � 11.88 ln x �36.94

x

x

Model It

x g’s

0.2 158

0.4 80

0.6 53

0.8 40

1.0 32

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118. Data Analysis An object at a temperature of 160 C wasremoved from a furnace and placed in a room at 20 C.The temperature of the object was measured each hour

and recorded in the table. A model for the data is givenby The graph of this model isshown in the figure.

(a) Use the graph to identify the horizontal asymptote ofthe model and interpret the asymptote in the contextof the problem.

(b) Use the model to approximate the time when thetemperature of the object was 100 C.

Synthesis

True or False? In Exercises 119–122, rewrite each verbalstatement as an equation. Then decide whether thestatement is true or false. Justify your answer.

119. The logarithm of the product of two numbers is equal tothe sum of the logarithms of the numbers.

120. The logarithm of the sum of two numbers is equal to theproduct of the logarithms of the numbers.

121. The logarithm of the difference of two numbers is equalto the difference of the logarithms of the numbers.

122. The logarithm of the quotient of two numbers is equal tothe difference of the logarithms of the numbers.

123. Think About It Is it possible for a logarithmic equationto have more than one extraneous solution? Explain.

124. Finance You are investing dollars at an annualinterest rate of compounded continuously, for years.Which of the following would result in the highest valueof the investment? Explain your reasoning.

(a) Double the amount you invest.

(b) Double your interest rate.

(c) Double the number of years.

125. Think About It Are the times required for the invest-ments in Exercises 107 and 108 to quadruple twice aslong as the times for them to double? Give a reason foryour answer and verify your answer algebraically.

126. Writing Write two or three sentences stating the generalguidelines that you follow when solving (a) exponentialequations and (b) logarithmic equations.

Skills Review

In Exercises 127–130, simplify the expression.

127.

128.

129.

130.

In Exercises 131–134, sketch a graph of the function.

131.

132.

133.

134.

In Exercises 135–138, evaluate the logarithm using thechange-of-base formula. Approximate your result to threedecimal places.

135.

136.

137.

138. log8 22

log3�4 5

log3 4

log6 9

g�x� � �x � 3,x2 � 1,

x ≤ �1x > �1

g�x� � �2x,�x2 � 4,

x < 0x ≥ 0

f �x� � x � 2 � 8

f �x� � x � 9

3�10 � 2

3�25 � 3�15

�32 � 2�25

�48x2y 5

tr,P

T

h

Tem

pera

ture

(in

degr

ees

Cel

sius

)

20

40

60

80

100

120

140

160

1 2 3 4 5 6 7 8

Hour

T � 20 �1 � 7�2�h��.h

T�

256 Chapter 3 Exponential and Logarithmic Functions

Hour, h Temperature, T

0

1

2

3

4

5 24�

29�

38�

56�

90�

160�

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