This document outlines three theorems for logarithms: 1) The Product Theorem states that the log of a product is equal to the sum of the logs of the factors. 2) The Quotient Theorem states that the log of a quotient is equal to the difference of the logs of the factors. 3) The Power Theorem states that the log of a factor to a given power is equal to the power times the log of the factor. These theorems only apply when the logarithms have the same base. Examples are provided to demonstrate applying the theorems to simplify logarithmic expressions and evaluate logarithms.

1. 1
A) Product Theorem
The log of a product of two factors is equal to the _____of the logs of the factors.
log(a)(b) log(a) log(b)
B) Quotient Theorem
The log of a quotient of two factors is equal to the _________ of the logs of the
factors.
a
log log(a) log(b)
b
C) Power Theorem
The log of a factor to a given power is equal to the power ______ the log of the
factor.
log(an ) n log(a)
All of the above theorems only apply to logs with the SAME BASE.
Examples
1. Write as a single log and then evaluate.
a. log 2 4 log 2 8
b. 4 log 2 x log 2 ( x 1) log 2 ( x 2 1)

2. 2
Expand the following:
x y2 x ( x4 )
3
A) log 3 B) log
x y2 x
Apply the Properties
Example:
Given that log8 2 0.33333 and that log8 3 0.52852 , find log8 18
Example:
Given log a 2 0.3562
10
log a 3 0.5646 Find the value of log a
3
log a 5 0.8271