Math 102
5.4 "Logarithmic Functions"
Objectives:
* Learn how to graph logarithmic functions.
De…nition of Logarithmic Function
De…nition:
"Logarithmic Function"
For b > 0 and b 6= 1; the logarithmic function with base b is denoted f (x) = logb (x) ,
where
y = logb (x)
if and only if by = x
NOTE: Two special logarithmic functions are:
(Natural logarithmic function)
f (x) = loge (x) = ln (x)
f (x) = log10 (x) = log (x)
(Common logarithmic function)
Example 1: (Evaluating logarithmic functions)
Find the indicated values.
a) log2
1
2
b) log2 (1)
c) log2 (2)
d) log2 (4)
Graphs of Logarithmic Functions
Example 2: (graphing a logarithmic function)
Graph f (x) = 2x and f (x) = log2 (x) on the same coordinate plane.
State the domain and range for each function.
y
y = 2x
x
1
2
(x; y)
x
1; 12
1
4
2
20
(0; 1)
1
1
1
(1; 2)
2
2
(2; 4)
4
2
2
6
(x; y)
1
2
0
2
y = log2 (x)
-2
2
-2
4
x
-4
Example 3: (graphing a decreasing logarithmic function)
Graph f (x) = log1=2 (x).
y
x
y
(x; y)
1
2
2
1
1
2
-2
2
-1
4
x
-2
Page: 1
Notes by Bibiana Lopez
College Algebra
5.4
Properties of Logarithmic Functions:
The function f (x) = logb x has the following properties:
1: If b > 1 then f (x) is:
2: x
If 0 < b < 1 then f (x) is:
intercept of the graph of f (x) is: _________
3: Vertical asymptote is: _________
4: Domain: __________
Range: __________
Example 4: (graphing logarithmic functions using transformations)
Graph f (x) = log2 (x + 2) : State the domain and range.
y
x
1
2
y
1
(x; y)
1
2;
(x; y)
4
1
1
0
(1; 0)
2
1
(2; 1)
6
2
-2
2
4
x
-2
-4
Example 5: (graphing logarithmic functions using transformations)
Graph f (x) = 4
log2 (x
3) : State the domain and range.
y
x
1
2
y
1
(x; y)
1
2;
(x; y)
(x; y)
(x; y)
8
6
1
4
2
1
0
(1; 0)
2
1
(2; 1)
5
10
-2
x
-4
Example 6: (graphing logarithmic functions using transformations)
Graph f (x) = 1 + log1=2 (x
1) : State the domain and range.
y
x
1
2
1
2
y
(x; y)
1
1
2;
0
(1; 0)
1
(x; y)
(x; y)
6
4
1
2
(2; 1)
-2
2
-2
4
x
-4
Page: 2
Notes by Bibiana Lopez