College Algebra 9th txtbk

SECTION 5–3 Logarithmic Functions 355
10
y
f
y 2 x
y x
x
f
y 2 x
x
f 1
y log 2 x
3
1
8
1
8
3
5
5
5 10
f 1
y log 2 x
x
2
1
1
4
1
2
0 1
1 2
1
4
1
2
2
1
1 0
2 1
2 4
4 2
5
3 8
8 3
DOMAIN of f ( , ) RANGE of f 1
RANGE of f (0, ) DOMAIN of f 1
Z Figure 1 Logarithmic function with base 2.
Ordered
pairs
reversed
MATCHED PROBLEM 1 Repeat Example 1 for f (x) ( 1 2) x and f 1 (x) log 1 2 x.
Z DEFINITION 1 Logarithmic Function
For b 7 0, b 1, the inverse of f (x) b x , denoted f 1 (x) log b x, is the
logarithmic function with base b.
Logarithmic form
y log b x
is equivalent to
Exponential form
x b y
The log to the base b of x is the exponent to which b must be raised to obtain x.
For example,
y
0 1
y log b x
0 b 1
x
y log 10 x is equivalent to
y log e x is equivalent to
Remember: A logarithm is an exponent.
x 10 y
x e y
y
0 1
DOMAIN (0, )
RANGE (, )
(a)
y log b x
b 1
DOMAIN (0, )
RANGE (, )
(b)
Z Figure 2 Typical logarithmic
graphs.
x
It is very important to remember that the equations y log and x b y
b x
define the same function, and as such can be used interchangeably.
Because the domain of an exponential function includes all real numbers and its range
is the set of positive real numbers, the domain of a logarithmic function is the set of all
positive real numbers and its range is the set of all real numbers. For example, log 10 3 is
defined, but log 10 0 and log 10 (5) are not defined.
In short, the function y log b x for any b is only defined for positive x values. Typical
logarithmic curves are shown in Figure 2. Notice that in each case, the y axis is a vertical
asymptote for the graph.
The graphs in Example 1 and Figure 2 suggest that logarithmic graphs share some common
properties. Several of these properties are listed in Theorem 1. It might be helpful in
understanding them to review Theorem 1 in Section 5-1. Each of these properties is a consequence
of a corresponding property of exponential graphs.