7INVERSE FUNCTIONSINVERSE FUNCTIONS

7.3Logarithmic Functions

INVERSE FUNCTIONS

In this section, we will learn about:

Logarithmic functions and natural logarithms.

If a > 0 and a ≠ 1, the exponential function

f(x) = ax is either increasing or decreasing,

so it is one-to-one.

Thus, it has an inverse function f -1, which

is called the logarithmic function with base a

and is denoted by loga.

LOGARITHMIC FUNCTIONS

If we use the formulation of an inverse

function given by (7.1.3),

then we have:

1 ( ) ( )f x y f y x

log ya x y a x

Definition 1LOGARITHMIC FUNCTIONS

Thus, if x > 0, then logax is the exponent

to which the base a must be raised

to give x.

LOGARITHMIC FUNCTIONS

Evaluate:

(a) log381

(b) log255

(c) log100.001

LOGARITHMIC FUNCTIONS Example 1

(a) log381 = 4 since 34 = 81

(b) log255 = ½ since 251/2 = 5

(c) log100.001 = -3 since 10-3 = 0.001

LOGARITHMIC FUNCTIONS Example 1

The cancellation equations (Equations 4

in Section 7.1), when applied to the functions

f(x) = ax and f -1(x) = logax, become:

log

log ( ) for every

for every 0

a

xa

x

a x x

a x x

LOGARITHMIC FUNCTIONS Definition 2

The logarithmic function loga has

domain and range .

It is continuous since it is the inverse of a continuous function, namely, the exponential function.

Its graph is the reflection of the graph of y = ax about the line y = x.

(0, )

LOGARITHMIC FUNCTIONS

The figure shows the case where

a > 1.

The most important logarithmic functions have base a > 1.

LOGARITHMIC FUNCTIONS

The fact that y = ax is a very rapidly

increasing function for x > 0 is reflected in the

fact that y = logax is a very slowly increasing

function for x > 1.

LOGARITHMIC FUNCTIONS

The figure shows the graphs of y = logax

with various values of the base a > 1.

Since loga1 = 0, the graphs of all logarithmic functions pass through the point (1, 0).

LOGARITHMIC FUNCTIONS

The following theorem

summarizes the properties

of logarithmic functions.

LOGARITHMIC FUNCTIONS

If a > 1, the function f(x) = logax is

a one-to-one, continuous, increasing

function with domain (0, ∞) and range . If x, y > 0 and r is any real number, then

PROPERTIES OF LOGARITHMS Theorem 3

1. log ( ) log log

2. log log log

3. log ( ) log

a a a

a a a

ra a

xy x y

xx y

y

x r x

Properties 1, 2, and 3 follow from the

corresponding properties of exponential

functions given in Section 7.2

PROPERTIES OF LOGARITHMS

Use the properties of logarithms

in Theorem 3 to evaluate:

(a) log42 + log432

(b) log280 - log25

Example 2PROPERTIES OF LOGARITHMS

Using Property 1 in Theorem 3,

we have:

This is because 43 = 64.

4 4 4

4

log 2 log 32 log 2 32

log 64

3

Example 2 aPROPERTIES OF LOGARITHMS

Using Property 2, we have:

This is because 24 = 16.

2 2 2

2

80log 80 log 5 log

5

log 16

4

Example 2 bPROPERTIES OF LOGARITHMS

The limits of exponential functions given

in Section 7.2 are reflected in the following

limits of logarithmic functions.

Compare these with this earlier figure.

LIMITS OF LOGARITHMS

If a > 1, then

In particular, the y-axis is a vertical asymptote of the curve y = logax.

LIMITS OF LOGARITHMS Equation 4

0lim log and lim loga ax x

x x

As x → 0, we know that t = tan2x → tan20 = 0and the values of t are positive.

Hence, by Equation 4 with a = 10 > 1, we have:

LIMITS OF LOGARITHMS Example 3

2100

Find lim log tan .x

x

210 100 0

lim log tan lim logx t

x t

Of all possible bases a for logarithms,

we will see in Chapter 3 that the most

convenient choice of a base is the number e,

which was defined in Section 7.2.

NATURAL LOGARITHMS

The logarithm with base e is called

the natural logarithm and has a special

notation:

log lne x x

NATURAL LOGARITHMS

If we put a = e and replace loge with ‘ln’

in (1) and (2), then the defining properties of

the natural logarithm function become:

ln yx y e x

Definitions 5 and 6NATURAL LOGARITHMS

ln(ex ) x x °

eln x x x 0

In particular, if we set x = 1,

we get:ln 1e

NATURAL LOGARITHMS

Find x if ln x = 5.

From (5), we see thatln x = 5 means e5 = x

Therefore, x = e5.

E. g. 4—Solution 1NATURAL LOGARITHMS

If you have trouble working with the ‘ln’

notation, just replace it by loge.

Then, the equation becomes loge x = 5.

So, by the definition of logarithm, e5 = x.

NATURAL LOGARITHMS E. g. 4—Solution 1

Start with the equation ln x = 5.

Then, apply the exponential function to both

sides of the equation: eln x = e5

However, the second cancellation equation in Equation 6 states that eln x = x.

Therefore, x = e5.

NATURAL LOGARITHMS E. g. 4—Solution 2

Solve the equation e5 - 3x = 10.

We take natural logarithms of both sides of the equation and use Definition 9:

As the natural logarithm is found on scientific calculators, we can approximate the solution—to four decimal places: x ≈ 0.8991

5 3ln( ) ln10

5 3 ln10

3 5 ln10

1(5 ln10)

3

xe

x

x

x

Example 5NATURAL LOGARITHMS

Express as a single

logarithm.

Using Properties 3 and 1 of logarithms, we have:

12ln lna b

1/ 212ln ln ln ln

ln ln

ln( )

a b a b

a b

a b

Example 6NATURAL LOGARITHMS

The following formula shows that

logarithms with any base can be

expressed in terms of the natural

logarithm.

NATURAL LOGARITHMS

For any positive number a (a ≠ 1),

we have:ln

loglna

xx

a

Formula 7CHANGE OF BASE FORMULA

Let y = logax.

Then, from (1), we have ay = x.

Taking natural logarithms of both sides of this equation, we get y ln a = ln x.

Therefore,ln

ln

xy

a

ProofCHANGE OF BASE FORMULA

Scientific calculators have a key for

natural logarithms.

So, Formula 7 enables us to use a calculator to compute a logarithm with any base—as shown in the following example.

Similarly, Formula 7 allows us to graph any logarithmic function on a graphing calculator or computer.

NATURAL LOGARITHMS

Evaluate log8 5 correct to six

decimal places.

Formula 7 gives: 8

ln 5log 5 0.773976

ln8

Example 7NATURAL LOGARITHMS

The graphs of the exponential function y = ex

and its inverse function, the natural logarithm

function, are shown.

As the curve y = ex crosses the y-axis with a slope of 1, it follows that the reflected curve y = ln x crosses the x-axis with a slope of 1.

NATURAL LOGARITHMS

In common with all other logarithmic functions

with base greater than 1, the natural

logarithm is a continuous, increasing function

defined on and the y-axis is

a vertical asymptote.

(0, )

NATURAL LOGARITHMS

If we put a = e in Equation 4,

then we have these limits:

NATURAL LOGARITHMS Equation 8

0lim ln lim lnx x

x x

Sketch the graph of the function

y = ln(x - 2) -1.

We start with the graph of y = ln x.

NATURAL LOGARITHMS Example 8

Using the transformations of Section 1.3, we shift it 2 units to the right—to get the graph of y = ln(x - 2).

Example 8NATURAL LOGARITHMS

Then, we shift it 1 unit downward—to get the graph of y = ln(x - 2) -1.

Notice that the line x = 2 is a vertical asymptote since:

NATURAL LOGARITHMS Example 8

2

lim ln 2 1x

x

We have seen that ln x → ∞ as x → ∞.

However, this happens very slowly.

In fact, ln x grows more slowly than any positive power of x.

NATURAL LOGARITHMS

To illustrate this fact, we compare

approximate values of the functions

y = ln x and y = x½ = in the table.x

NATURAL LOGARITHMS

We graph the functions here.

Initially, the graphs grow at comparable rates. Eventually, though, the root function far surpasses

the logarithm.

NATURAL LOGARITHMS

In fact, we will be able to show in

Section 7.8 that:

for any positive power p.

So, for large x, the values of ln x are very small compared with xp.

NATURAL LOGARITHMS

lnlim 0

px

x

x