Section 4.2
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
y  log a x = x  a
y
• Evaluate the following…
f ( x)  log2 32
f ( x)  log4 2
1
f ( x)  log10 100
f ( x)  log3 1
TI Cares!
• Calculator Applications: the common log
(log button on the calculator) uses the
common base, 10.
f ( x)  log10 10
f ( x)  log
f ( x)  log10 2.5
f ( x)  log10  2
1
10 3
Basic Logarithmic Properties
Examples: log8 8  1
log6 1  0
Examples: log 7 7  2
2
5log5 8  8
The voice balloons should help you see the
"undoing" that takes place between the
exponential and logarithmic functions in the
inverse properties.
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
Find…
ln e
3
log a a
log 7 1
2
log2 16
Entrance Slip
1) Write the exponent in log form
4  64
3
3
2
25  125
2) Evaluate the following
log 9 3
1
log 4
4
log 2 16
Graphs of logarithmic
Functions
Example
Use transformations to graph g(x)=2+log 3 ( x  3).
Start with log 3 x.
y






x


























Example
Use transformations to graph g(x)=-log3 x
y






x


























Example
Use transformations to graph g(x)=log3 ( x)
y






x


























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) = log 4 ( x  5)
Common Logarithms
Example
I
Use the formula R=log to solve the problem. If an
I0
earthquake is 100 times as intense as a zero-level quake
(I=100 I 0 ), what is its magnitude on the Richter Scale?
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. e ln 5
Example
y  log 3 x  2
4
2
-5
5
-2
-4
-6
y   log 6 x  2
Example
4
2
-5
5
-2
-4
-6
y  log 5 x  2  3
Example
4
2
-5
5
-2
-4
-6
1)
y  log 4 x
2)
y  ln( x  4)
3)
y  ln x  3
4)
y  2  ln x
5)
y  log( x  5)
6)
y  3  log( x  2)
7)
y  3 log x
8)
y  2  log( x  1)
9)
y  2  log 2 x
10)
y  1  log 5 ( x  6)
Graph the following…
1)
2)
3)
4)
5)
6)
7)
8)
Exit Slip
Graph the following logarithmic equations.
1) log 5 x
2) 6  log 4 x
3) 4  log 6 x  5
4) log x  3  1