Chapter 5 Exponential, Logarithmic, and Inverse Trigonometric Functions

Chapter 5 Exponential, Logarithmic, and Inverse Trigonometric Functions

Exponential Functions

If f (x) = x2, then the base x varies and the exponent 2 is a constant. The function f (x) = 2x has constant base 2 and variable exponent x. This is called an exponential

function.

x f (x) = 2x

-3 1/8

-2 1/4

-1 1/2

0

1

1

2

2

4

3

8

Now lim 2x = and lim 2x = 0

x

x-

To see that f (x) = 2x gets large quickly consider the following:

Suppose you have one penny and for the entire month of February your money doubles

each day. If I offer you $100,000 or the amount you would have after February 28, which would you take?

Day

Amount

Dollars

1

2

$0.02

7

27 = 1,024

$10.24

14

214 = 16,384

$163.84

21 221 = 2,097,152 $20,971.52

28 228 = 2.6844 ? 108 $2,684,400

Hence you can see how fast the exponential function rises.

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Now let f (x) =

1 2

x = 2-x

x 2-x -2 4 -1 2 01 1 1/2 2 1/4

The most important exponential function is f (x) = ex. We base our development on the following definition:

lim(1 + x)1/x = e

x0

x 1

10 1

100 1

1,000 1

10,000 1

100,000

(1 + x)1/x (1.1)10 = (1.01)100 = (1.001)1,000 = (1.0001)10,000 = (1.00001)100,000 =

Do the above calculations on your calculator and find that e 2.7182818.

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Now let f (x) = ex and complete the table of values on your calculator.

x ex 0 1 10 100 1000 -1 -10 -100

Thus the graph of f (x) = ex is

lim ex = and lim ex = 0

x

x-

If 0 < a, a = 1, then f (x) = ax is the general exponential function. All of the familiar laws of exponents hold. If 1 < a, then y = ax is increasing and if 0 < a < 1, then y = ax is decreasing. Thus, if ax1 = ax2, then x1 = x2. Also note that ax > 0.

129

Exercises on Exponential Functions

1. Graph

a. f (x) = 3x

b. f (x) =

1x 3

c. f (x) = e-x

d. f (x) = -(2-x)

2. Find the limits

a. lim (1.001)x

x

b. lim (.999)x

x

c. lim e-x

x

d. lim e-x

x-

e.

lim

x

e2x e2x

- +

e-2x e-2x

f. lim e1/(1-x)

x1+

g. lim e1/(1-x)

x1-

3. Suppose the graphs of f (x) = x2 and g(x) = 2x are drawn where the unit of

measurement is 1 in. If x = 24 in, then find f (x) in feet and g(x) in miles.

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Logarithmic Functions

Definition: loga x = y means ay = x. loga x is read logarithm of x to the base a.

Example 1

Let y = log2 x

x log2 x 10 21 42 83 1/2 -1 1/4 -2

Remarks: log2(0) = y means 2y = 0 which is impossible. Also log2(-1) = y means 2y = -1 which is impossible. Thus the domain of y = log2 x is (0, ).

Common Logarithm

The common logarithm has base a = 10. We write y = log10 x = log x which means

10y

=

x.

Thus

log 1

=

0,

log 10

=

1,

log 100

=

2,

log

1 10

=

-1,

etc.

Natural Logarithm

The natural logarithm has base a = e. We write y = loge x = ln x which means ey = x. Complete the following table on your calculator.

x ln x

1

e e2

10

1000

1 e

1 1000

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