Coburn J.W., Herdlick J.D. College Algebra: Graphs and Models

Подождите немного. Документ загружается.

81. 82.

54321⫺5⫺4⫺3⫺2⫺1

⫺1

⫺2

⫺3

⫺4

⫺5

54321

⫺5⫺4⫺3⫺2⫺1

⫺1

⫺2

⫺3

⫺4

⫺5

For the functions given, (a) ﬁnd (x), then use your calculator to verify they are inverses by (b) using ordered pairs,

85. 86.

87. 88. j1x2⫽ 22x ⫹ 9

⫺ 6h1x2⫽

x ⫹ 1

g1x2⫽ x

⫹ 1; x ⱖ 0f 1x2⫽ 2x ⫹ 1

y ⴝ x

ⴚ1

490 CHAPTER 5 Exponential and Logarithmic Functions 5–12

College Algebra Graphs & Models—

83. 84.

54321

⫺5⫺4⫺3⫺2⫺1

⫺1

⫺2

⫺3

⫺4

⫺5

54321

⫺5⫺4⫺3⫺2⫺1

⫺1

⫺2

⫺3

⫺4

⫺5

䊳

WORKING WITH FORMULAS

89. The height of a projected image: f(x)

The height of an image projected on a screen

is given by the formula shown, where

represents the actual height of the image on the

projector (in centimeters) and x is the distance of

the projector from the screen (in centimeters).

(a) When the projector is 80 cm from the screen,

how large is the image? (b) Show that the inverse

function is then input your

answer from part (a) and comment on the result.

What information does the inverse function

give?

⫺1

1x2⫽ 2x ⫹ 17,

f 1x2

ⴝ

x ⴚ 8.5

90. The radius of a sphere: r(x)

In generic form, the radius of a sphere is given by

the formula shown, where r(x) represents the radius

and x represents the volume of the sphere in cubic

units. (a) If a weather balloon that is roughly

spherical holds 14,130 in

of helium, what is the

radius of the balloon (use ? (b) Show that

the inverse function is then input

your answer from part (a) and comment on the

result. What information does the inverse function

give?

⫺1

1x2⫽

␲x

␲ ⬇ 3.142

ⴝ

4␲

䊳

APPLICATIONS

91. Temperature and altitude: The temperature (in degrees Fahrenheit) at a given altitude can be approximated by

the function where f(x) represents the temperature and x represents the altitude in thousands of

feet. (a) What is the approximate temperature at an altitude of 35,000 ft (normal cruising altitude for commercial

airliners)? (b) Find and state what the independent and dependent variables represent. (c) If the

temperature outside a weather balloon is , what is the approximate altitude of the balloon?

92. Fines for speeding: In some localities, there is a set formula to determine the amount of a ﬁne for exceeding

posted speed limits. Suppose the amount of the ﬁne for exceeding a 50 mph speed limit was given by the

function ( ) where f(x) represents the ﬁne in dollars for a speed of x mph. (a) What is

the ﬁne for traveling 65 mph through this speed zone? (b) Find and state what the independent and

dependent variables represent. (c) If a ﬁne of $172 were assessed, how fast was the driver going through this

speed zone?

93. Effect of gravity: Due to the effect of gravity, the distance an object has fallen after being dropped is given by

the function where f(x) represents the distance in feet after x sec. (a) How far has the object

fallen 3 sec after it has been dropped? (b) Find and state what the independent and dependent variables

represent. (c) If the object is dropped from a height of 784 ft, how many seconds until it hits the ground (stops

falling)?

⫺1

1x2,

1x2⫽ 16x

; x ⱖ 0,

⫺1

1x2,

x 7 50f

1x2⫽ 12x ⫺ 560

⫺18°F

⫺1

1x2,

1x2⫽⫺

x ⫹ 59,

cob19545_ch05_479-492.qxd 11/30/10 3:52 PM Page 490

5–13 Section 5.1 One-to-One and Inverse Functions 491

College Algebra Graphs & Models—

94. Area and radius: In generic form, the area of a

circle is given by where f(x) represents

the area in square units for a circle with radius x.

(a) A pet dog is tethered to a stake in the backyard.

If the tether is 10 ft long, how much area does the

dog have to roam (use )? (b) Find

and state what the independent and dependent

variables represent. (c) If the owners want to allow

the dog 1256 ft

of area to live and roam, how long

a tether should be used?

95. Volume of a cone: In

generic form, the volume

of an equipoise cone

(height equal to radius) is

given by

where f(x) represents

the volume in units

and

x represents the height of the cone. (a) Find the

volume of such a cone if ft (use

(b) Find and state what the independent andf

1

1x2,

␲ ⬇ 3.142.r  30

1x2

␲x

1

1x2,␲ ⬇ 3.14

1x2 ␲x

dependent variables represent. (c) If the volume of

water in the cone is 763.02 ft

, how deep is the

water at its deepest point?

96. Wind power: The power delivered by a certain

wind-powered generator can be modeled by the

function where f(x) is the horsepower

(hp) delivered by the generator and x represents

the speed of the wind in miles per hour. (a) Use

the model to determine how much horsepower is

generated by a 30 mph wind. (b) The person

monitoring the output of the generators (wind

generators are usually erected in large numbers)

would like a function that gives the wind speed

based on the horsepower readings on the gauges

in the monitoring station. For this purpose, ﬁnd

and state what the independent and dependent

variables represent. (c) If gauges show 25.6 hp is

being generated, how fast is the wind blowing?

1

1x2

f 1x2

2500

h  r

䊳

EXTENDING THE CONCEPT

97. For a deeper understanding of the algebraic method

for ﬁnding an inverse, suppose a function f is

deﬁned as . We can then

deﬁne the inverse as ,

having interchanged x and y in the equation

portion. The equation for is not in standard

form, but (x, y) still represents all ordered pairs

satisfying either equation. Solving for y gives

, and demonstrates the

role of steps 2, 3, and 4 of the method. (a) Find ﬁve

ordered pairs that satisfy the equation for f, then

(b) interchange their coordinates and show they

satisfy the equation for .f

1

:e1x, y2 ` y 

 2f

1

: 51x, y2 | x  3y  66

f 1x2:51x, y2 | y  3x  66

98. By inspection, which of the following is the inverse

function for ?

d. f

1

1x2

ax 

b

1

1x2

ax 

b

1

1x2

1x  22

1

1x2

ax 

b

1x2

ax 



䊳

MAINTAINING YOUR SKILLS

99. (R.3) Write as many of the following formulas as you can from memory:

a. perimeter of a rectangle d. volume of a cone g. area of a trapezoid

b. area of a circle e. circumference of a circle h. volume of a sphere

c. volume of a cylinder f. area of a triangle i. Pythagorean theorem

102. (R.4) Solve the following cubic equations by

factoring:

d. x

 3x

 4x  0

 3x

 0

 7x

 4x  28  0

 5x  0

100. (3.2) Given solve the

inequality using the x-intercepts and end-

behavior of the graph.

101. (3.4) For the function ﬁnd the

average rate of change between and

and between and Which is greater?

Why?

x  5.x  4

x  2,x  1

y  21x  3

f1x2  0

f 1x2 x

 x  2,

cob19545_ch05_479-492.qxd 9/2/10 9:24 PM Page 491

492 CHAPTER 5 Exponential and Logarithmic Functions 5–14

College Algebra Graphs & Models—

Demographics is the statistical study of human populations. In this section, we intro-

duce the family of exponential functions, which are widely used to model population

growth or decline with additional applications in science, engineering, and many other

ﬁelds. As with other functions, we begin with a study of the graph and its characteristics.

A. Evaluating Exponential Functions

In the boomtowns of the old west, it was not

uncommon for a town to double in size every year

(at least for a time) as the lure of gold drew more

and more people westward. When this type of

growth is modeled using mathematics, exponents

play a lead role. Suppose the town of Goldsboro

had 1000 residents when gold was ﬁrst discovered.

After 1 yr the population doubled to 2000 resi-

dents. The next year it doubled again to 4000, then

again to 8000, then to 16,000 and so on. You probably recognize the digits in blue as

powers of two (indicating the population is doubling), with each one multiplied by 1000

(the initial population). This suggests we can model the relationship using

where P(x) is the population after x yr. Further, we can evaluate this function, called

an exponential function, for fractional parts of a year using rational exponents. The

population of Goldsboro one-and-a-half years after the gold rush was

In general, exponential functions are deﬁned as follows.

Exponential Functions

For , , and all real numbers x,

deﬁnes the base b exponential function.

Limiting b to positive values ensures that outputs will be real numbers, and the

restriction is needed since is a constant function (1 raised to any power

is still 1). Speciﬁcally note the domain of an exponential function is all real numbers,

and that all of the familiar properties of exponents still hold. A summary of these prop-

erties follows. For a complete review, see Section R.2.

Exponential Properties

For real numbers a, b, m, and n, with a, ,

n

 a

n



1ab2

 a

 b

mn

 b

mn

b 7 0

y  1

b  1

1x2 b

b  1b 7 0

⬇ 2828 people

 1000

1122

P a

b 1000

P1x2 1000

LEARNING OBJECTIVES

In Section 5.2 you will see

how we can:

A. Evaluate an exponential

function

B. Graph general

exponential functions

C. Graph base-e

exponential functions

D. Solve exponential

equations and

applications

5.2 Exponential Functions

WORTHY OF NOTE

To properly understand the

exponential function and its graph

requires that we evaluate

even when x is irrational. For

example, what does mean?

While the technical details require

calculus, it can be shown that

successive approximations of

as in 2

2.2360

, 2

2.23606

, 2

2.236067

, . . .

approach a unique real number,

and that exists for all real

numbers x.

f 1x2 2

492 5–14

cob19545_ch05_479-492.qxd 9/2/10 9:24 PM Page 492

5–15 Section 5.2 Exponential Functions 493

College Algebra G&M—

EXAMPLE 1

䊳

Evaluating Exponential Functions

Evaluate each exponential function for , , , and . Use a

calculator for , rounding to ﬁve decimal places.

a. b.

Solution

䊳

a. For , b. For ,

Now try Exercises 7 through 10

䊳

B. Graphing Exponential Functions

To gain a better understanding of exponential functions, we’ll graph examples of

and note some of the characteristic features. Since , it seems reasonable

that we graph one exponential function where and one where 0 6 b 6 1.b 7 1

b  1y  b

g1␲2 a

␲

 0.07827 f 1␲2 4

␲

 77.88023

b a



f a

b 4

 14  2

g112 a

1



112 4

1



g122 a



122 4

 16

g1x2 a

f 1x2 4

g1x2 a

f 1x2 4

x  ␲

x  ␲x 

x 1x  2

A. You’ve just seen how we

can evaluate an exponential

function

EXAMPLE 2

䊳

Graphing Exponential Functions with b  1

Graph using a table of values.

Solution

䊳

To get an idea of the graph’s shape we’ll use integer values from to 3 in our

table, then draw the graph as a continuous curve, since the function is deﬁned for

all real numbers.

3

y  2

(0, 1)

(1, 2)

(2, 4)

(3, 8)

44

3 2

 8

 4

 2

 1

1



1

2



2

3



3

y ⴝ 2

Now try Exercises 11 and 12

䊳

Several important observations can now be made. First note the x-axis (the line

) is a horizontal asymptote for the function, because as Second,

the function is increasing over its entire domain, giving the function a range of

y 僆 10, q2.

y S 0.x S q,y  0

WORTHY OF NOTE

As in Example 2, functions that are

increasing for all are said to

be monotonically increasing or

simply monotonic functions. The

function in Example 3 is monotoni-

cally decreasing.

x 僆 D

cob19545_ch05_493-503.qxd 9/2/10 9:26 PM Page 493

494 CHAPTER 5 Exponential and Logarithmic Functions 5–16

College Algebra G&M—

EXAMPLE 3

䊳

Graphing Exponential Functions with 0 ⬍ b ⬍ 1

Graph using a table of values.

Solution

䊳

Using properties of exponents, we can write as Again using

integers from to 3, we plot the ordered pairs and draw a continuous curve.3

x

 2

x

y  1

Now try Exercises 13 and 14

䊳

We note this graph is also asymptotic to the x-axis, but decreasing on its domain.

In addition, both and have a y-intercept of (0, 1) and both are

one-to-one, which suggests that an inverse function can be found. Finally, observe that

is a reﬂection of across the y-axis, a property that indicates these basic

graphs might also be transformed in other ways, as were the toolbox functions. The

characteristics of exponential functions are summarized here:

y  b

x

y  2

x

 1

y  2

3 2

3



2



1



 1

112

 2

 21

122

 2

 42

132

 2

 83

y ⴝ 2

ⴚx

(0, 1)

(1, 2)

(2, 4)

(3, 8)

44

and

• one-to-one function • y-intercept (0, 1)

• domain: • range:

• increasing if • decreasing if

• asymptotic to the x-axis (the line )

Just as the graph of a quadratic function maintains its parabolic shape regardless

of the transformations applied, exponential functions will also maintain their gen-

eral shape and features. Any sum or difference applied to the basic function

will cause a vertical shift in the same direction as the sign, and

any change to input values vs. will cause a horizontal shift in a

direction opposite the sign. For cases where multiple transformations are to be

applied, refer to the sequence outlined on page 209.

y  b

21y  b

xh

1y  b

 k vs. y  b

(0, 1)

(1, b)

f(x)  b

b  1

44

(0, 1)

0  b  1

44

f(x)  b

 

Figure 5.18 Figure 5.19

y  0

0 6 b 6 1b 7 1

y 僆 10, q2x 僆⺢

b ⴝ 1f

1x2ⴝ b

, b ⬎ 0

WORTHY OF NOTE

When an exponential function is

increasing, it can be referred to as a

“growth function.” When decreasing,

it is often called a “decay function.”

Each of the graphs shown in

Figures 5.18 and 5.19 should now

be added to your repertoire of

basic functions, to be sketched

from memory and analyzed or used

as needed.

cob19545_ch05_493-503.qxd 11/27/10 12:26 AM Page 494

5–17 Section 5.2 Exponential Functions 495

College Algebra G&M—

EXAMPLE 4

䊳

Graphing Exponential Functions Using Transformations

Graph using transformations of

the basic function (not by simply plotting

points). Clearly state what transformations are

applied.

Solution

䊳

The graph of F is that of the basic function

with a horizontal shift 1 unit right and

a vertical shift 2 units up. With this in mind the

horizontal asymptote also shifts from to

and (0, 1) shifts to (1, 3). The y-intercept of

F is at (0, 2.5):

To help sketch a more accurate graph, the point (3, 6) can be used:

Now try Exercises 15 through 30

䊳

C. The Base-e Exponential Function: f(x) ⴝ e

In nature, exponential growth occurs when the rate of change in a population’s growth

is in constant proportion to its current size. Using the rate of change notation,

where k is a constant. For the city of Goldsboro, we know the population at

time t is given by , but have no information on this value of k (see

Exercise 90). We can actually rewrite this function, and other exponential functions,

using a base that gives the value of k directly and without having to apply the differ-

ence quotient. This new base is an irrational number, symbolized by the letter e. In

Section 5.6 we’ll develop the number e in the context of compound interest, while

making numerous references to our discussion here, where we deﬁne e as follows.

The Number e

For

In words, e is the number that approaches as x becomes inﬁnitely large.

It has been proven that as x grows without bound, indeed approaches the

unique, irrational number that we have named e. Table 5.5 gives approximate values of

the expression for selected values of x, and shows to ﬁve decimal places.e ⬇ 2.71828

a1 

as x Sq, a1 

S e

x 7 0,

P1t2 1000

¢P

¢t

 kP,

F132 6.

 2.5



 2

 2

1

 2

F102 2

1021

 2

y  2

y  0

1x2 2

f 1x2 2

F1x2 2

x1

 2

44

(1, 3)

(3, 6)

(0, 2.5)

F(x) = 2

is shifted

1 unit right

2 units up

y  2

B. You’ve just seen how

we can graph general

exponential functions

WORTHY OF NOTE

Just as the ratio of a circle’s

circumference to its diameter is an

irrational number symbolized by ,

the irrational number that results

from for inﬁnitely large

x is symbolized by e. Writing

exponential functions in terms of e

simpliﬁes many calculations in

advanced courses, and offers

additional advantages in applications

of exponential functions.

a1 

␲

cob19545_ch05_493-503.qxd 11/30/10 3:55 PM Page 495

496 CHAPTER 5 Exponential and Logarithmic Functions 5–18

College Algebra G&M—

The result is the base-e exponential

function: , also called the natural

exponential function. Instead of having to

enter a decimal approximation when comput-

ing with e, most calculators have an “e

” key,

usually as the function for the key

marked . To ﬁnd the value of e

, use the

keystrokes 2 , and the calcu-

lator display should read 7.389056099. Note

the calculator supplies the left parenthesis for

the exponent, and you must supply the right.

See Figure 5.20.

EXAMPLE 5

䊳

Evaluating the Natural Exponential Function

Use a calculator to evaluate for the values of x given. Round to six

decimal places as needed.

a. f(3) b. f(1) c. f(0) d.

Solution

䊳

a. b.

c. (exactly) d.

Now try Exercises 31 through 36

䊳

Although e is an irrational number, the graph

of behaves in exactly the same way and has

the same characteristics as other exponential graphs.

Figure 5.21 shows this graph on the same grid as

and As we might expect, all three

graphs are increasing, have an asymptote at

and contain the point (0, 1), with the graph of

“between” the other two. The domain for all three

functions, as with all basic exponential functions, is

with range The same

transformations applied earlier can also be applied to

the graph of See Exercises 37 through 42.

D. Solving Exponential Equations Using

the Uniqueness Property

Since exponential functions are one-to-one, we can solve equations where each side is

an exponential term with the identical base. This is because one-to-oneness guarantees

a unique solution to the equation.

Exponential Equations and the Uniqueness Property

For all real numbers m, n, and b, where and

1. If 2. If

then then

Equal bases imply exponents are equal.

 b

m  n.

m  n,b

 b

b  1,b 7 0

y  e

y 僆 10, q2.x 僆 1q, q2

y  e

y  0,

y  3

.y  2

y  e

f 1

2 e

 1.648721f 102 e

 1

112 e

 2.718282f 132 e

 20.085537

1x2 e

ENTER

)

LN2nd

2nd

f 1x2 e

Figure 5.20

Approximate Value

10 2.59

100

2.705

1000 2.7169

10,000

2.71815

100,000 2.718268

1,000,000

2.7182805

10,000,000 2.71828169

11 ⴙ

Table 5.5

543215 4 3 2 1

1

2

y  2

y  3

y  e

Figure 5.21

C. You’ve just seen

how we can graph base-e

exponential functions

WORTHY OF NOTE

Exponential functions are very

different from the power functions

studied earlier. For power functions,

the base is variable and the

exponent is constant: while

for exponential functions the

exponent is a variable and the base

is constant: y  b

y  x

cob19545_ch05_493-503.qxd 9/2/10 9:27 PM Page 496

5–19 Section 5.2 Exponential Functions 497

College Algebra G&M—

The equation can be written as and we note is a solution.

Although can be written as the bases are not alike and the solution to

this equation must wait until additional tools are developed in Section 5.5.

EXAMPLE 6

䊳

Solving Exponential Equations

Solve the exponential equations using the uniqueness property.

a. b. c.

Solution

䊳

a. given

rewrite using base 3

uniqueness property

solve for

Check

䊳

given

substitute for

simplify

result checks

The remaining checks are left to the student.

given

rewrite using base 6

power property of exponents

uniqueness property

solve for

c. given

product property; quotient property

simplify

uniqueness property

add

, subtract 2

solve for

Now try Exercises 43 through 62

䊳

Earlier in this section, we showed the exponential function was deﬁned

for all real numbers and was a one-to-one function. This is important because it estab-

lishes that equations like must have a solution, even if x is not rational. In fact,

since and the following inequalities indicate the solution must be

between 2 and 3

7 is between 4 and 8

replace 4 with 2

, 8 with 2

must be between 2 and 3

Until we develop an inverse for exponential functions, we are unable to solve many

of these equations in exact form. We can, however, get a very close approximation using

a graphing calculator. For the equation enter and on the

screen. Then press 6 to graph both functions (see Figure 5.22). To ﬁnd the point of

intersection, press (CALC) and select option 5:intersect, then press three

ENTER

TRACE

2nd

ZOOM

 7Y

 2

 7,

1 2 6 x 6 3

6 2

4 6 7 6 8

 8,2

 4

 7

1x2 b

x 

2 x  1

1 x  2  3  x

x2

 e

3x

x2

 e

4 1x12



x1

x  0

1 3x  2  2x  2

3x2

 6

2x2

1

3x2

 16

x1

3x2

 36

x1

81  81

 81

51

 81

21

 81

2x1

 81

x 

1 2x  1  4

2x1

 3

2x1

 81



x1

3x2

 36

x1

2x1

 81

 2

 32

x  52

 2

 32

cob19545_ch05_493-503.qxd 9/2/10 9:27 PM Page 497

times (to identify the intersecting functions and

bypass “Guess”). The x- and y-coordinates of the

point of intersection will appear at the bottom of

the screen, with the x-coordinate being the solu-

tion. As you can see, x is indeed between 2 and 3.

See Exercises 63 through 66.

Two common applications of exponential

functions involve appreciation (as when an item

grows in value over time), and depreciation (as

when tools and equipment decrease in value

over time).

EXAMPLE 7

䊳

Applications of Exponential Functions—Depreciation

For insurance purposes, it is estimated that large household appliances lose of

their value each year. The current value can then be modeled by the function

where V

is the initial value and V(t) represents the value after t

years. How many years does it take a washing machine that cost $625 new, to

depreciate to a value of $256?

Solution

䊳

For this exercise, and The formula yields

given

substitute known values

divide by 625

equate bases

Uniqueness Property

After 4 yr, the washing machine’s value has dropped to $256.

Now try Exercises 69 through 74

䊳

Another very practical application of the natural exponential function involves

Newton’s law of cooling. This law or formula models the temperature of an object as

it cools down, as when a pizza is removed from the oven and placed on the kitchen

counter. The function model is

where T

represents the initial temperature of the object, T

represents the temperature

of the room or surrounding medium, T(x) is the temperature of the object xmin later, and

k is the cooling rate as determined by the nature and physical properties of the object.

EXAMPLE 8

䊳

Applying an Exponential Function—Newton’s Law of Cooling

A pizza is taken from a oven and placed on the counter to cool. If the

temperature in the kitchen is , and the cooling rate for this type of pizza is

a. What is the temperature (to the nearest degree) of the pizza 2 min later?

b. To the nearest minute, how long until the pizza has cooled to a temperature

below ?

c. If Zack and Raef like to eat their pizza at a temperature of about , how

many minutes should they wait to “dig in”?

110°F

90°F

k 0.35

75°F

425°F

T1x2 T

 1T

 T

, k 6 0

1 4  t

256

625

 a

256

625

 a

256  625a

V1t2 V

V1t2 $256.V

 $625

V1t2 V

498 CHAPTER 5 Exponential and Logarithmic Functions 5–20

College Algebra G&M—

Figure 5.22

1010

10

cob19545_ch05_493-503.qxd 9/2/10 9:27 PM Page 498

Solution

䊳

Begin by substituting the given values to obtain the equation model:

general equation model

substitute 75 for

, 425 for

, and ⴚ0.35 for

simplify

For part (a) we simply ﬁnd T(2):

substitute 2 for

result

Two minutes later, the temperature of the pizza is near .

b. Using the TABLE feature of a graphing calculator shows the pizza reaches a

temperature of just under after 9 min: .

c. We elect to use the intersection-of-graphs

method. After setting an appropriate

window, we enter

and , then press option

5:intersect. After pressing three

times, the coordinates of the point of

intersection appear at the bottom of the

screen: It appears the

boys should wait about min for the

pizza to cool.

Now try Exercises 75 and 76

䊳

There are a number of additional applications of exponential functions in the

Exercise Set. See Exercises 77 through 84.

y  110.x  6.6,

ENTER

CALC

2nd

 110

 75  350e

0.35

T192 90°F90°

249°

 249

T122 75  350e

0.35122

 75  350e

0.35x

 75  1425  752e

0.35x

T1x2 T

 1T

 T

5–21 Section 5.2 Exponential Functions 499

College Algebra G&M—

6. Discuss/Explain the statement, “For the

y-intercept of is ”10, a  k2.y  ab

 k

k 7 0,

䊳

CONCEPTS AND VOCABULARY

Fill in each blank with the appropriate word or phrase. Carefully reread the section if needed.

1. An exponential function is one of the form

, where

and is any real number.

 1,7 0,

y 

2. The domain of is all ,

and the range is . Further, as

, y .

x S q

y 僆

y  b

3. For exponential functions of the form the

y-intercept is (0, ), since

for any real number b.



y  ab

5. State true or false and explain why: is

always increasing if 0 6 b 6 1.

y  b

4. If each side of an equation can be written as an

exponential term with the same base, the equation

can be solved using the .

100

400

5.2 EXERCISES

D. You’ve just seen how

we can solve exponential

equations and applications

cob19545_ch05_493-503.qxd 11/27/10 12:26 AM Page 499