Math 35 (Spring ’09)
9.5 "Logarithmic Functions"
Objectives:
*
De…ne, evaluate and graph logarithmic functions.
*
Write logarithmic equations as exponential equations.
*
Write exponential equations as logarithmic equations.
Preliminaries:
In this section, we will discuss inverses of exponential functions. These functions are called logarithmic functions, and
they can be used to solve problems from …elds such as: electronics, seismology (the study of earthquakes), and business.
De…nition of Logarithm
For all positive numbers b; where b 6= 1; and all positive numbers x,
Example 1: (Write logarithmic equations as exponential equations)
Write each logarithmic equation as an exponential equation:
a)
log4 64 = 3
b)
log6
Example 2: (Write exponential equations as logarithmic equations)
Write each exponential equation as a logarithmic equation:
b)
a)
9 1 = 19
1
=
36
1 2
4
=
2
1
16
Example 3: (Finding the value of x by writing logarithmic equations as exponential equations)
a)
log1=3 x = 2
b)
log3 81 = x
Example 4: (Evaluating logarithmic expressions)
Evaluate each logarithmic expression:
a)
log9 81
Example 5: (Using the notation log x = log10 x)
Evaluate each logarithmic expression, if possible:
1
a)
log 10
b)
log 10
b)
c)
Page: 1
log4
1
16
log ( 10)
d)
log 0
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
9.5
Graph Logarithmic Functions
Logarithmic Functions
If b > 0 and b 6= 1; the logarithmic function with base b is de…ned by the equations
Example 6: (Graphing logarithmic functions)
Graph each function.
a) f (x) = log2 x
y
x
f (x)
or
b) g (x) = log1=2
y
6
x
4
g (x)
4
2
-6
-4
6
2
-2
2
4
6
-2
-6
x
-4
-2
2
4
6
-2
-4
-4
-6
-6
x
Properties of Logarithmic Functions:
The function f (x) = logb x has the following properties:
1: If b > 1 then the function f (x) is :
If 0 < b < 1 then the function f (x) is :
2: x
intercept of the graph of f (x) is
3: Vertical asymptote is:
4: Domain:
Range:
Example 7: (Graphing logarithmic functions)
Graph each function.
a) f (x) = log1=3 (x + 2)
y
x
f (x)
b) g (x) = (log2 (x
3)) + 1
y
6
x
4
g (x)
4
2
2
-6
-4
-2
2
-2
6
4
6
x
-6
-4
-2
2
-4
-4
-6
-6
Page: 2
4
-2
Notes by Bibiana Lopez
6
x