1. Properties of Logarithms
Module 2 Day 2
Properties of Logarithms

2. Product Property log 𝑏 (𝑥·𝑦) = log 𝑏 𝑥 + log 𝑏 𝑦 or
Example:
log 2 (7𝑥) = log log 2 𝑥
log log = log 2 (10)

3. Quotient Property log 𝑏 (𝑥/𝑦) = log 𝑏 𝑥 − log 𝑏 𝑦 or
Example:
log 2 ( 7 𝑥 ) = log 2 7 − log 2 𝑥
log 2 6 − log = log 2 (3)

4. Power Property log 𝑏 (𝑥) 𝑝 = p( log 𝑏 𝑥 ) Or p( log 𝑏 𝑥 ) = log 𝑏 𝑥 𝑝
Example:
log 2 (𝑥) 3 = 3 log 2 𝑥
3 log 2 𝑥 = log 2 (𝑥) 3

5. Property of Equality If log 𝑏 (𝑥) = log 𝑏 (𝑦) , 𝑡ℎ𝑒𝑛 x = y Example:
If there is one log on each side and the bases are the same, the logs can cancel
Example:
If log 𝑏 (𝑥) = log 𝑏 (2) , then x = 2
If log 𝑏 (𝑥+7) = log 𝑏 (2) , 𝑡ℎ𝑒𝑛 𝑥+7=2 or x =-5

6. If given specific values of logs:
Use the properties of logs to break down the number into values that are given.
Substitute the given values into the logs
Evaluate and simplify
Example: Given that log 6 7= log 6 3= log 6 2=.39 evaluate the following
log 6 21
Answer: log 6 3·7= log log 6 7
= 1.69
log 6 18
Answer: log 6 3² ∙2= 2log log 6 2
2(.61) = 1.61
log
Answer: log 6 3² 6 = 2log 6 3 − log 6 6
2(.61) – 1 = .22