3.2 Logarithmic Functions and their Graphs

Students will recognize and evaluate logarithmic functions with base a.

Students will graph logarithmic functions.

Students will recognize, evaluate, and graph natural logarithmic functions.

Students will use logarithmic functions to model and solve real-life problems.

Logarithm Conversion

If then

is called the common logarithm, it means

is called the natural logarithm, it means

y xblog b xy

y xlogy xlog10

y xlny xelog

Example 1

Use the logarithm conversion to evaluate each logarithm at the indicated value of x.

a. b.

c. d.

y x x log ,2 32 y x x log ,3 1

y x x log ,4 2 y x x log ,10

1

100

Example 2

Use a calculator to evaluate the function at each value of x.

a. x=10 b. x=2.5 c. x=-2 d.

f x x( ) log 10

x 1

4

Properties of Logarithms1.

2.

3.

4.

loga 1 0

log ( ) log logb b buv u v

log log logb b b

u

vu v

log logbx

bu x u

Example 3

Solve for x.

a. b.

Simplify.

c.

log log2 2 3x log4 4 x

log5 5x

Example 4

In the same coordinate plane, sketch the graph of each function by hand.

a. b. f x x( ) 2 g x x( ) log 2

y

x–2

2

Example 5

Each of the following graphs is a transformation of

a. b.

y

x–2

2

f x x( ) log 10

g x x( ) log ( ) 10 1 h x x( ) log 2 10

Example 6Use a calculator to evaluate the function

where:a. b. c.

f x x( ) lnx 2 x .3 x 1

Example 7

Use the properties of natural logarithms to rewrite each expression.

a) b)

c) d)

ln1

elne

lne0 2 lne

Example 8

Find the domain of each function

a. f(x) = ln (x – 2) b. g(x) = ln (2 – x)