2.5 Laws of Logarithms

Law for Logarithms of Products
log a
mn log a
m log a
n
The log of a product of factors equals the sum of the logs of the factors.
Use the law of logarithms of products to evaluate log 6
4 log 6
9.
log 6
4 log 6
9 log 6
(4)(9) Simplify.
log 6
36 Evaluate.
2
Now find the difference between logs with the same base.
Example 2
Subtracting Logarithms with the Same Base
Express log a
m log a
n as a single logarithm.
Solution
Let log a
m x and let log a
n y. Therefore, log a
m log a
n x y.
Also, since log a
m x, a x m. Since log a
n y, a y n.
Therefore,
m n
a x
a
y
m n ax y
Rewrite m n ax y in logarithmic form.
log a m n x y However, from above, log a m x and log a n y.
So log a m n log a m log a n.
Law for Logarithms of Quotients
log a m n log a m log a n, n ≠ 0
The log of a quotient equals the log of the dividend less the log of the
divisor.
Use the law for logarithms of quotients to simplify log 2
18 log 2
9.
log 2
18 log 2
9 log 2 1 8
9 Simplify.
log 2
2
1
Evaluate.
122 CHAPTER 2 EXPONENTIAL AND LOGARITHMIC FUNCTION MODELS