# Summary: Logarithmic Properties

## 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, 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

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**Provided by:**OpenStax**Authored by:**Jay Abramson, et al..**Located at:**https://openstax.org/books/precalculus/pages/1-introduction-to-functions.**License:**CC BY: Attribution.**License terms:**Download For Free at : http://cnx.org/contents/[email protected].. - College Algebra.
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