1
f
Section 5.3 Logarithmic Functions
Objective 1: Understanding the Definition of a Logarithmic Function
Every exponential function of the form
( ) x
f x b
where
0b
and
1b
is one-to-one and thus has an
inverse function.
The graph of
( ) , 1
x
f x b b
and its inverse.
To find the equation of
1
f
:
Step 1. Change
( )f x
to y:
x
y b
Step 2. Interchange x and y:
y
x b
Step 3. Solve for y:??
Before we can solve for y we must introduce the following definition:
Definition of the Logarithmic Function
For
0, 0 and 1x b b
, the logarithmic function with base b is defined by
logb
y x
if and only if
y
x b
.
Step 3. Solve for y:
y
x b
can be written as
logb
y x
Step 4. Change y to
1( )f x
:
1( ) logb
f x x
5.3.1
Write the exponential equation as an equation involving a logarithm.
5.3.9
Write the logarithmic equation as an exponential equation.
( )
x
f x b
(0,1)
(1, )b
1
( 1, )
b
(1, 0)
1
( , 1)
b
( ,1)b
