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]

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, 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.

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

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