Summary: Logarithmic Properties

Key Equations

The Product Rule for Logarithms	${\mathrm{log}}_{b}\left(MN\right)={\mathrm{log}}_{b}\left(M\right)+{\mathrm{log}}_{b}\left(N\right)$
The Quotient Rule for Logarithms	${\mathrm{log}}_{b}\left(\frac{M}{N}\right)={\mathrm{log}}_{b}M-{\mathrm{log}}_{b}N$
The Power Rule for Logarithms	${\mathrm{log}}_{b}\left({M}^{n}\right)=n{\mathrm{log}}_{b}M$
The Change-of-Base Formula	${\mathrm{log}}_{b}M\text{=}\frac{{\mathrm{log}}_{n}M}{{\mathrm{log}}_{n}b}\text{ }n>0,n\ne 1,b\ne 1$

Key Concepts

We can use the product rule of logarithms to rewrite the log of a product as a sum of logarithms.
We can use the quotient rule of logarithms to rewrite the log of a quotient as a difference of logarithms.
We can use the power rule for logarithms to rewrite the log of a power as the product of the exponent and the log of its base.
We can use the product rule, the quotient rule, and the power rule together to combine or expand a logarithm with a complex input.
The rules of logarithms can also be used to condense sums, differences, and products with the same base as a single logarithm.
We can convert a logarithm with any base to a quotient of logarithms with any other base using the change-of-base formula.
The change-of-base formula is often used to rewrite a logarithm with a base other than 10 and e as the quotient of natural or common logs. That way a calculator can be used to evaluate.

Glossary

change-of-base formula a formula for converting a logarithm with any base to a quotient of logarithms with any other base.

power rule for logarithms a rule of logarithms that states that the log of a power is equal to the product of the exponent and the log of its base

product rule for logarithms a rule of logarithms that states that the log of a product is equal to a sum of logarithms

quotient rule for logarithms a rule of logarithms that states that the log of a quotient is equal to a difference of logarithms

Exponential and Logarithmic Equations and Models