4.1 EXPONENTS AND EXPONENTIAL FUNCTIONS - Utah State University

[Pages:15]196

Chapter 4 Exponential and Logarithmic Functions

4.1 E X P O N E N T S A N D E X P O N E N T I A L F U N C T I O N S

What I really am is a mathematician. Rather than being remembered as the first woman this or that, I would prefer to be remembered, as a mathematician should, simply for the theorems I have proved and the problems I have solved.

Julia Robinson

Elementary algebra courses define expressions of the form b x for integer exponents (and a few rational-number exponents). We need to expand this to allow more kinds of numbers as exponents. This requires extending definitions to nth roots and then to rational exponents. The extension to irrational exponents is properly left to calculus, but we can at least get a feeling for what a calculator does when we evaluate an expression such as 32 or 2. In the following definitions, n and m denote positive integers.

Definition: exponents, roots, and radicals

Integer Exponents Principle nth Root Rational Exponents

b n b ? b ? . . . ? b, product of n factors, if n 0 b 0 1 if b 0; bn 1b n, n 0 and b 0. b 1n n b is the real number root of x n b when there is only one root; when there are two, b 1n is the positive root. When n 2 we write b 12 b.

If mn is in lowest terms, then b mn b 1nm. When b 0, b mn is also equal to n b m, which is called radical form.

Irrational Exponents

Certain theoretical considerations require care in defining a number like 22 but properties of the real number system guarantee its existence. We use calculators to evaluate exponential expressions. Since

2 1.41421356 . . . ,

we would expect the numbers 21.4, 21.41, 21.414, . . . (where all of the exponents are rational to approach 22. The calculator makes the conclusion plausible:

21.4 275 2.639 22 2.6651441.

21.41 2.6574

21.414 2.66475

Properties of Exponents

In the expression b x we call b the base and x the exponent. If b is a positive number, then b x is a real number for every value of x. If, however, b is negative, then b x is a real number for some values of x, but it is nonreal for other values of x. For instance, 453 is a real number (see Example 2b), but 432 is a nonreal complex number. Our primary interest in this chapter is the exponential function, which requires a positive base. Therefore, the following properties of exponents assume b and c are positive.

4.1 Exponents and Exponential Functions

197

Properties of exponents

If b and c are positive numbers and x and y are any real numbers, then

E1. b xb y b xy

E2.

bx by

b xy

E3. b xy b xy

E4. bcx b xc x

E5.

b c

x

bx cx

.

TECHNOLOGY TIP Roots of negative numbers

Different calculators handle roots of negative numbers differently. Check to see how your calculator evaluates 113. We know that 1 is the only real root of x 3 1, so 113 1. Your calculator may use yx or .

57 57 Remember parentheses for both 1 and the 13. If the display returned is 1, then your calculator evaluates 113 as you expect. Your calculator may display an ERROR message (which means that your machine does not evaluate roots of negative numbers), or you may get something like (.5, .866 . . .), which means your calculator is giving you a complex number root. When b is negative and the exponent is irrational, do not expect a real number result.

If your calculator doesn't return what you expect, you have to be more clever than your calculator. Remember that cube roots of negative numbers are defined and that

b13 b 13.

To make certain that you know how to use the above definitions and to get your calculator to evaluate exponential expressions, make certain that you can do everything suggested in the first example.

EXAMPLE 1 Exponential expressions Simplify and evaluate (in exact form if possible, five-decimal place approximation otherwise):

(a) 3 64 (b) 423 (c) 853 (d) 42

Solution

(a) 3 64 6413 6413 2613 22 4. We use the calculator to check by evaluating 6413.

(b) 423 can be rewritten in other forms, as, for example, 4213 3 16, but other equivalent forms are no easier to evaluate. In decimal form, 4^23 2.51984 (be careful about parentheses).

(c) 853 2353 25 32. (d) 42 7.10299.

EXAMPLE 2 Calculator evaluation Give a four-decimal place approximation to illustrate E2 and E3.

(a) 52 and 552 Solution

(b) 5 ? 2 and 52

(a) Evaluating 52 and rounding off to four decimal places gives 16.1208. Evaluating 5 and 52 and then dividing, also returns 16.1208.

(b) Rounding to four decimal places, both 5 ? 2 and 52 are given by the

calculator as 1274.7996.

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Chapter 4 Exponential and Logarithmic Functions

Strategy: (a) Use E4 first,

followed by E3, and sim-

plify. (b) First replace x2

by

1 x2

and

4 x1

by

4 x

,

then

simplify.

EXAMPLE 3 Getting rid of negative exponents Simplify. Express the result without negative exponents.

(a) x2y 32

x2 4x1 5 (b) 5x 1

Solution

(a)

x2y 32

x22y 32 x 4y6 x 4

1 y6

x4 y6

.

(b)

x2

4x1 5x 1

5

1 x2

4 x

5

5x 1

1 4x 5x2

x2 5x 1

x

x2

1 .

1

5x1 x 25x

1

x

Strategy: Multiply numerator and denominator by x 1 and simplify to get rid of the radical in the denominator.

Strategy: First get rid of the negative exponents by multiplying both sides by x 2, then solve the resulting quadratic equation.

EXAMPLE 4 Rationalize denominator Rationalize the denominator of x1 .

x 1

Solution Follow the strategy.

x1 x 1

x 1x 1 x 1x 1

x

1x x1

1

x

1.

EXAMPLE 5 Disguised quadratic equation Solve the equation 2x2 7x1 4 0.

Solution Follow the strategy.

x 22x2 7x1 4 x 2 ? 0 2 7x 4x 2 0

Factoring, 2 x1 4x 0. By the zero-product principle, the solutions are

2

and

1 4

.

Strategy: First express 27 3 9

as a power of 3, then use properties of exponents.

EXAMPLE 6

Equating powers of 3

Solve the

equation

32x1

27 3 9

0.

Solution Follow the strategy.

27 3 9

33 3 32

33 323

3323

373

Therefore, the given equation is equivalent to

32x1 373.

In this form it is intuitively clear that the two exponents must be equal:

2x

1

7 3

.

Thus

the

solution

is

2 3

.

If we had been unable to express 27 as a simple power of 3, then the solution

3 9

of this problem would have had to await the techniques of Section 4.4.

4.1 Exponents and Exponential Functions

199

y

Rational Power Functions, x myn

(? 1, 1)

y = x6

y = x4 y = x2

(1, 1) x

(0, 0)

(a) Even powers y

1

y = x62

1

y = x64

(1, 1)

1

y = x66

In Chapter 3, our focus was on polynomial functions, which can all be expressed as sums of power functions, x 0, x 1, x 2, x 3, . . . . With the definition of rational

exponents, it makes sense to consider graphs of rational powers of x, functions of the form f x x mn, where m and n are positive integers (for a negative ex-

ponent, we would take the reciprocal). We have already looked at the graph of y x x 12, which we recognize as the inverse of the function f x x 2, x 0.

There is a basic difference between graphs of even and odd powers of x. The even powers form a family, all of whose graphs contain the points (1, 1), (0, 0), and (1, 1). See Figure 1a. As the power increases, the graphs become progressively

flatter around the origin and then increase more and more steeply, as if a slightly

flexible parabola had been " jammed nose first" into the x- axis. None of these even powers is one-one, but each is increasing if we restrict the domain to x 0. Thus for even numbers n, restricting the domain to the nonnegative real numbers gives a function y x n, x 0 with an inverse function y x 1n, x 0. See Figure 1b.

The odd powers of x also form a family. All graphs contain the points (1, 1), (0, 0), (1, 1). All odd power functions are increasing and hence oneone. Therefore every odd power function y x n has an inverse function y x 1n that is also increasing, and the domain (and range) for every member of the family,

including inverses, consists of all real numbers. See Figure 2.

x

(0, 0)

y

(b) Even roots FIGURE 1

y = x7

y = x5 y = x3

(?1, ?1)

(1, 1)

x (0, 0)

y

1

y = x65

1

y = x63

(1, 1)

1

y = x67 x

(0, 0)

(?1, ?1)

(a) Odd powers

FIGURE 2

(b) Odd roots

From the definition, x mn x 1nm. Thus to graph y x 23, we enter Y X132 . The parentheses are critical to make sure that the calculator is graphing what we intend. The graphs of several rational power functions are shown in Figure 3 and are typical of such functions in general. The variations in shape depend on the parity (odd or even) of m and n. Rather than trying to describe all possible combinations, we suggest that you experiment and observe the patterns, being careful with parentheses. See the following Technology Tip.

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Chapter 4 Exponential and Logarithmic Functions

2

y = x63

4

y = x63

[? 3, 3] by [? 1.5, 2.5] (a)

3

y = x65

[? 3, 3] by [? 1.5, 2.5] (b)

2

y = x65

[? 3, 3] by [? 1.5, 2.5]

[? 3, 3] by [? 1.5, 1.5]

(c)

(d)

FIGURE 3

TECHNOLOGY TIP What your calculator may not show you

To graph rational power functions correctly, you need to be sure that you

know how your calculator handles such expressions. The graph of y x 23

is shown in Figure 3a. The function is defined for all real numbers x and

has what is called a "cusp" at the origin, a sharp corner at a local minimum.

Without care, your calculator will almost surely not duplicate the graph in

Figure

3a.

If

you

enter

Y

X23 ;

you

will

probably

get

the

parabola

y

1 3

x

2.

Graphing Y X(23), you will get a function whose domain is the set of

nonnegative numbers. To get the graph in Figure 3a, you will probably have

to enter Y (X(13))2 . On the HP-38 and HP-48, even that function will

produce only the right half of the graph, the points where x 0. You simply

must recognize that the graph of the function contains more than the

calculator shows in that case.

EXAMPLE 7 A shifted rational power function Describe the graph of f x 1 x 223 in terms of basic transformations of a rational power function. For what values of x is f increasing? Find all local extrema.

Solution If gx x 23, then the graph of y x 223 is a horizontal shift of the graph of g, 2 units right, and y x 223 is a reflection of the shifted graph through the

(2, 1)

2

y = 1 ? (x ? 2)3 [? 3, 6] by [? 3, 6]

FIGURE 4

4.1 Exponents and Exponential Functions

201

x-axis (tipping it upside down). Finally, the graph of f is obtained by shifting up 1 unit.

Graphing Y 1 ((X 2(13))2 gives a picture something like Figure 4. It is clear that f is increasing on , 2 and that there is a local maximum at (2, 1). Because gx x 23 has a minimum at the origin, f has only the one local extremum.

Beyond Calculator Precision

There are times when we need more precision than a calculator can display. If we understand some basic principles, we may be able to do more than the calculator alone can provide. The idea of one?one functions has some unexpected applications that are used a number of times in this chapter. For example, suppose a2 b 2. What can we say about a and b? Because two numbers can have the same square (as 22 22, without more information, all we can say is that a b. If, however, a2 b 2 and we know that both a and b are positive, then we can conclude that a b. We are using the fact that the function y x 2 is a one?one function on the limited domain where x 0. We use this idea in the next example.

EXAMPLE 8 Do equal decimals imply equality? Which, if any, of the following are equal?

a 5 1

b 5 21 45

c 5702887 1762289

Solution When we evaluate the three numbers by calculator, each shows the same display, 3.2360679775, so relying on the calculator alone, we would have to conclude that the numbers are equal. Their appearance is so different, though, that we want more confirmation.

For the first pair, a and b, we can get rid of some of the radicals by squaring.

a2 6 25 b 2 5 21 45

These numbers still appear very different, but rather than squaring again immediately, we observe that it would be much easier to square b 2 5, so we subtract 5 from each and then square again.

a2 52 1 252 1 45 20 21 45. b 2 52 21 45.

Since a2 52 b 2 52 and a2 5, b 2 5 are both positive, we have

a2 5 b 2 5, so a2 b 2, and finally, since a and b are positive, a b.

Now, how about a and c? Since c is clearly a rational number, we might be able

to use the technique of Example 7 from Section 3.3 to show that a is an irrational

number. As an alternative, we use an approach that shows how to go beyond the

number of digits a calculator can display. We begin with the idea of squaring. Before

squaring, though, we subtract 1 from both a and c, and then we can clear fractions.

That

is,

we

want

to

know

if

a

1

c

1,

or

if

5

, 3940598

1762289

and

then

if

176228952 39405982. What the calculator shows for both is 1.55283125976E13.

That is, the display tells us only that each number equals 155283125976??; the last

two digits are not displayed. Here we use what we know about properties of

multiplication. While we cannot display the entire number, we can use the calcula-

tor for either the first digits or the last.

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Chapter 4 Exponential and Logarithmic Functions

39405982 ends . . . 5982 . . . 7604, and 17622892 ? 5 ends . . . 28925 . . . 7605.

Putting the information together, we have

176228952 15528312597605, and 39405982 15528312597604.

We conclude that a c, so that a and b are equal to each other, but c is different from either a or b.

Exponential Functions

We assume the properties of real numbers that assure us that for any real number x, the expression 5x is a positive real number, so the equation f x 5x defines a positive-valued function whose domain is R, called an exponential function. The number 5 is the base of this exponential function, but any other positive number (except 1) can be used as a base for an exponential function as well.

Definition: exponential function

An exponential function, base b, is any function that can be expressed in the form

y

f x b x

8 7 6 5 4 3 2 1

?4 ?3 ?2 ?1

y = 3x

y = 2x (0, 1)

x 1234

FIGURE 5 Exponential functions with

bases greater than 1

y

8

7 y = ( 1)x

6

3

5

4

3

y = (0.5)x 2 (0, 1) 1

?4 ?3 ?2 ?1

x 1234

where b is a fixed positive number 1).

Graphs of Exponential Functions

Graphs of all exponential functions have one of essentially two different shapes,

depending on whether b 1 or b 1.

To get a feeling for the graphs when b 1, we use the graphing calculator to

graph exponential functions for two different bases. See Figure 5. The graphs in

Figure 5 are drawn on the same axes, but we suggest that you graph each of the

same functions, preferably on different screens, or at least sequentially, to see how

similar they are. If the base b is a number near 1, then the curve is relatively flat;

as b increases, the curve y b x rises more and more steeply to the right of the

y-axis.

The graphs of exponential functions when the base is a number less than 1 are

reflections through the y-axis of the kinds of curves in Figure 5. For example, if

f x

3x,

then

for

gx

1 3

x,

we

have

gx 31x 3x f x,

so the graph of g is the reflection through the y-axis of the graph of f. See Figure 6 and graph a variety of such functions yourself. Again, when the base b is a number near 1, the exponential curve is flatter, becoming steeper to the left of the y-axis as b decreases toward 0.

Properties of Exponential Functions

FIGURE 6 Exponential functions with

bases less than 1

The graphs in Figures 5 and 6 suggest some general properties of exponential functions.

4.1 Exponents and Exponential Functions

203

Properties of exponential functions

Suppose b is a positive number different from 1 and f x b x.

Domain: (, ) Range: 0, Intercepts: x-intercept points, none; y-intercept point (0, 1). Asymptotes: the x-axis is always a horizontal asymptote. If b 1, then f is an increasing function; if b 1, then f is a decreasing function.

Every exponential function f is one-one and thus has an inverse function.

The Euler Number e and the Natural Exponential Function

It turns out, as we shall see, that in one very important sense, all exponential functions can be considered as transformations of a single exponential function. That being the case, we should be able to choose any particular exponential function and use it as the exponential function, from which we can obtain all others. As a matter of fact, however, nature has made a selection for us. There is an important number just a little less than 3, denoted by e, which is the base of what is almost universally called the natural exponential function, and denoted by

f x ex or f x expx.

y

y = ex

8 7

y = 3x 6 5

(1, 3) 4 3

2

y = 2x (1, e) (1, 2)

1 (0, 1)

?4 ?3 ?2 ?1

x 123 4

FIGURE 7 Natural exponential function,

y ex

Justification for the name "natural" usually comes in a calculus course; for our

purposes, we simply state that all sorts of natural growth and decay phenomena are most easily described in terms of ex.

The number e, sometimes called the Euler number, can be defined in many different ways (see the Historical Note, " and e," Part I) and appears in as many unexpected mathematical contexts as the number . Your calculator is programmed to evaluate ex for real numbers x. The number itself is an irrational

number that has been calculated to many decimal places, the first twenty-five of

which are given by

e 2.71828 18284 59045 23536 02875.

You should see what your calculator displays by evaluating e1 or EXP (1).

Since e is a number between 2 and 3, and much nearer 3, we would expect the

graph of natural exponential function to lie between the graphs of y 2x and

y 3x, closer to the latter, as can be seen in Figure 7. You should be able to draw

a calculator graph similar to Figure 7, using the built-in function key for ex, which

is paired on almost all calculators with the LN key. Use a decimal window and 57

trace along the curves to the point where x 1. Compare the y-coordinates at that

point. On the natural exponential function, you should see part of what your

calculator displays for e.

One of the ways to define the number e is as the limit of the function

f x

1

1 x

x

as

x

increases

without

bound.

That

is,

lim1

xA

1 x

x

e.

The

expression

1

1 x

x

appears

when

we

compute

compound

interest

on

investments.

See "Compound Interest" formula in Section 4.5. We show how the graph of

y

1

1 x

x

is

related

to

the

number

e

in

the

next

example.

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