Section 3.2
Logarithmic Functions
The Definition of Logarithmic
Functions
To change from logarithmic form to the more
familiar exponential form, use the pattern;
y=log b x means b y  x
Example
Write each equation in the equivalent exponential form.
a. 4=log 2 x
b. x=log3 9
Example
Write each equation in its equivalent logarithmic form.
a. b 4  16
b. 52  x
Example
Evaluate.
a. log 3 81
b. log 36 6
c. log 51
Basic Logarithmic Properties
Examples: log8 8  1
log 61  0
Examples: log 7 7  2
2
5log5 8  8
Example
Use the properties of logarithms to find the answers.
a. 3log3 15
b. log 2 23
c. log 9 9
d. log
1
3 3
Graphs of logarithmic
Functions
The Domain of a Logarithmic
Function
We learned that the domain of an exponential function
of the form f(x)=b x includes all real numbers and its
range is the set of positive real numbers. Because the
logarithmic function reverses the domain and the range
of the exponential function, the domain of a logarithmic
function of the form f(x)=log b x is the set of all positive
real numbers. In general, the domain of f(x)=log b g ( x)
consists of all x for which g(x)>0.
Example
Find the domain of f(x) = log4 ( x  5)
Common Logarithms
Graphing Calculator
Natural Logarithms
The logarithmic function with base e is called the natural
logarithmic function. The function f(x)=log e x is usually
expressed as f(x)=ln x. Like the domain of all logarithmic
functions, the domain of the natural logarthmic function
f(x)=ln x is the set of all positive real numbers. Thus the
domain of f(x)= ln g(x) consists of all x for which g(x)>0.
Example
Evaluate.
a. ln e 2
b. eln 5
Evaluate log6 36
(a)
(b)
(c)
(d)
1
2
2
1

2
2
What is the domain for log5 ( x  4)
(a) x>-4
(b) x>4
(c) x>0
(d) x  -4