2.5 Laws of Logarithms

It is often necessary to use the laws of logarithms to solve logarithmic equations.
Example 5
Solving a Logarithmic Equation
Solve for x in log 3
x 1 2 log 3 36 log 3 18 log 3 2.
Solution
Express the right side as a single logarithm with base 3.
log 3
x 1 2 log 3 36 log 3 18 log 3
2 Use the law for logarithms of powers.
1
2
log 3
x log 3
(36) log 3
18 log 3
2 Simplify.
log 3
x log 3
6 log 3
18 log 3
2 Use the law for logarithms of
products.
log 3
x log 3
(6)(18) log 3
2 Use the law for logarithms of
quotients.
log 3
x log 3
(6)( 18)
Simplify.
2
log 3
x log 3
54
∴ x 54
Equivalent Logarithms
If log a
m log a
n, then m n, provided a > 0, and a ≠ 1.
This result is true only when the logarithms have the same base.
CHECK, CONSOLIDATE, COMMUNICATE
1. Evaluate log 8
16 log 8
4.
2. Evaluate log 2000 log 2.
3. Express 2 log 3
5 3 log 3
2 as a single logarithm.
4. Give an example to show that the law for logarithms of powers works.
KEY IDEAS
• If m and n are positive numbers, a is a positive number other than 1,
and p is any real number, then the following laws hold true:
Law for Logarithms of Products: log a
(mn) log a
m log a
n
Law for Logarithms of Quotients: log a m n log a m log a n, n ≠ 0
Law for Logarithms of Powers: log a
m p p log a
m
• If log a
m log a
n, then m n, provided a > 0, and a ≠ 1.
124 CHAPTER 2 EXPONENTIAL AND LOGARITHMIC FUNCTION MODELS