Summary: Properties of Logarithms

Key Equations

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

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, quotient rule, and 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 or [latex]e[/latex] as the quotient of natural or common logs. A calculator can then be used to evaluate it.

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