Section 4.3

Logarithmic Functions and Graphs

Flashback

Consider the graph of the exponential function y = f(x) = 3x.

• Is f(x) one-to-one?• Does f(x) have an inverse that is a function?• Find the inverse.

Inverse of y = 3x

f (x) = 3x

y = 3x

x = 3y

x = 3y

Now, solve for y.

y= the power to which 3 must be raised in order to obtain x.

x = 3y

Solve for y.

y= the power to which 3 must be raised in order to obtain x.

Symbolically, y = log 3 x

“The logarithm, base 3, of x.”

Logarithm

For all positive numbers a, where a 1,

Logax is an exponent to which the base a

must be raised to give x.

yaa =x isequivalent to y=logx

Logarithmic Form

Exponential Form

a exponenlog x y t

base

ya

exponent

a e

x

b s

All a log is . . . is an exponent!

Argument(always positive)

Logarithmic Functions• Logarithmic functions are inverses of

exponential functions.

Graph: x = 3y or y = log 3 x 1. Choose values for y.2. Compute values for x.3. Plot the points and connect them with a

smooth curve.

* Note that the curve does not touch or cross

the y-axis.

Logarithmic Functions continued

(1/27, 3)31/27

(1/9, 2)21/9

(1/3, 1)11/3

2

1

0

y

(9, 2)9

(3, 1)3

(1, 0)1

(x, y)x = 3y

Graph: x = 3y

y = log 3 x

Side-by-Side Comparison

f (x) = 3x f (x) = log 3 x

Comparing Exponential and Logarithmic Functions

Logarithmic Functions

• Remember: Logarithmic functions are inverses of exponential functions.

x

- 1a

The inverse of f(x) = a

is given by

f (x)=log x

Asymptotes

• Recall that the horizontal asymptote of the exponential function y = ax is the x-axis.

• The vertical asymptote of a logarithmic function is the y-axis.

ay=log x

Logarithms

• Convert each of the following to a logarithmic equation.a) 25 = 5x b) ew = 30

log

A logarithm is an exponent!

ya x y x a

Example

• Convert each of the following to an exponential equation.

• a) log7 343 = 3

log7 343 = 3 7 3 = 343

b) logb R = 12

The logarithm is the exponent.

The base remains the same.

Finding Certain Logarithms• Find each of the following.

a) log2 16 b) log10 1000

c) log16 4 d) log10 0.001

Common Logarithm

10

For allpositivenumbersx,

logx=log x

•Log button on your calculator

is the common log *

Logarithms, base 10, are called common logarithms.

Example• Find each of the following common

logarithms on a calculator. Round to four decimal places.

a) log 723,456b) log 0.0000245c) log (4)

Does not existERR: nonreal anslog (4)

4.61084.610833916log 0.0000245

5.85945.859412123log 723,456

RoundedReadoutFunction Value

Natural Logarithms•Logarithms, base e, are called

natural logarithms.

• The abbreviation “ln” is generally used for natural logarithms.

• Thus, ln x means loge x.

* ln button on your calculator

is the natural log *

Example• Find each of the following natural logarithms

on a calculator. Round to four decimal places.

a) ln 723,456b) ln 0.0000245c) ln (4)

Does not existERR: nonreal ansln (4)

10.616810.61683744ln 0.0000245

13.491813.49179501ln 723,456

RoundedReadoutFunction Value

Changing Logarithmic Bases

•The Change-of-Base FormulaFor any logarithmic bases a and b,

and any positive number M,

loglog .

loga

ba

MM

b

Use change of base formula when you have a logarithm that is not base 10 or e.

Example

Find log6 8 using common logarithms.

Solution: First, we let a = 10, b = 6, and M = 8. Then we substitute into the change-of-base formula:

10

16

0

loglog

l

8

og

1.1606

68

ExampleWe can also use base e for a conversion.

Find log6 8 using natural logarithms.

Solution: Substituting e for a, 6 for b and 8 for M, we have

6 6

loglog

log

ln81.1606

ln 6

88 e

e

Properties of Logarithms

a

a

For a>0, a 1

2.

For the logarithmic base e,

3.

loga=1

log 1=0

ln e=1

l 04. n1=

1.

Graphs of Logarithmic Functions

• Graph: y = f(x) = log6

x.– Select y.– Compute x.

21/36

11/6

3216

236

16

01

yx, or 6 y

Example• Graph each of the following. • Describe how each graph can be obtained

from the graph of y = ln x. • Give the domain and the vertical

asymptote of each function.

• a) f(x) = ln (x 2)

• b) f(x) = 2 ¼ ln x

• c) f(x) = |ln (x + 1)|

Graph f(x) = ln (x 2)• The graph is a shift 2 units

right.

• The domain is the set of all real numbers greater than 2.

• The line x = 2 is the vertical

asymptote.

1.0995

0.6934

03

0.6932.5

1.3862.25

f(x)x

Graph f(x) = 2 ¼ ln x• The graph is a vertical

shrinking, followed by a reflection across the x-axis, and then a translation up 2 units.

• The domain is the set of all positive real numbers.

• The y-axis is the vertical asymptote.

14

1.5985

1.7253

21

2.1730.5

2.5760.1

f(x)x

Graph f(x) = |ln (x + 1)|• The graph is a translation 1

unit to the left. Then the absolute value has the effect of reflecting negative outputs across the x-axis.

• The domain is the set of all real numbers greater than 1.

• The line x = 1 is the vertical asymptote.

1.9466

1.3863

0.6931

00

0.6930.5

f(x)x

Application: Walking Speed

• In a study by psychologists Bornstein and Bornstein, it was found that the average walking speed w, in feet per second, of a person living in a city of population P, in thousands, is given by the function

w(P) = 0.37 ln P + 0.05.

Application: Walking Speed continued

The population of Philadelphia, Pennsylvania, is 1,517,600. Find the average walking speed of people living

in Philadelphia.

Since 1,517,600 = 1517.6 thousand, we substitute 1517.6 for P, since P is in

thousands:

w(1517.6) = 0.37 ln 1517.6 + 0.05 2.8 ft/sec.

The average walking speed of people living in Philadelphia is about 2.8 ft/sec.