EXAMPLE
2
Converting from Logarithmic to Exponential Form
Write each logarithmic equation in exponential form.
Logarithmic
Equation
a.
log 10 100 = 2
b.
log 7 49 = 2
log 8 0.125 = -1
c.
Exponential
Form
The base of the logarithm becomes
the base of the power.
10 2 = 100
7 2 = 49
The logarithm is the exponent.
A logarithm can be a negative
number.
8 -1 = 0.125
d.
log 5 5 = 1
51 = 5
e.
log 12 1 = 0
12 0 = 1
Write each logarithmic equation in exponential form.
2a. log 10 10 = 1
-3
2b. log 12 144 = 2 2c. log _1 8 =
2
A logarithm is an exponent, so the rules for exponents also apply to logarithms.
You may have noticed the following properties in the last example.
Special Properties of Logarithms
For any base b such that b > 0 and b ≠ 1,
LOGARITHMIC FORM
EXPONENTIAL FORM
EXAMPLE
b1 = b
log 1010 = 1
10 1 = 10
b0 = 1
log 101 = 0
10 0 = 1
Logarithm of Base b
log bb = 1
Logarithm of 1
log b1 = 0
A logarithm with base 10 is called a common logarithm . If no base is written for
a logarithm, the base is assumed to be 10. For example, log 5 = log 10 5.
You can use mental math to evaluate some logarithms.
EXAMPLE
3
Evaluating Logarithms by Using Mental Math
Evaluate by using mental math.
A log 1000
10 ? = 1000
The log is the exponent.
10 3 = 1000
Think: What power of
the base is the value?
log 1000 = 3
250
1
B log 4 _
Chapter 4 Exponential and Logarithmic Functions
4
1
4? = _
4
1
4 -1 = _
4
1 = -1
log 4 _
4