Summary: Logarithmic Properties Key Equations The Product Rule for Logarithms logb(MN)=logb(M)+logb(N){\mathrm{log}}_{b}\left(MN\right)={\mathrm{log}}_{b}\left(M\right)+{\mathrm{log}}_{b}\left(N\right)logb(MN)=logb(M)+logb(N) The Quotient Rule for Logarithms logb(MN)=logbM−logbN{\mathrm{log}}_{b}\left(\frac{M}{N}\right)={\mathrm{log}}_{b}M-{\mathrm{log}}_{b}Nlogb(NM)=logbM−logbN The Power Rule for Logarithms logb(Mn)=nlogbM{\mathrm{log}}_{b}\left({M}^{n}\right)=n{\mathrm{log}}_{b}Mlogb(Mn)=nlogbM The Change-of-Base Formula logbM=lognMlognb n>0,n≠1,b≠1{\mathrm{log}}_{b}M\text{=}\frac{{\mathrm{log}}_{n}M}{{\mathrm{log}}_{n}b}\text{ }n>0,n\ne 1,b\ne 1logbM=lognblognM n>0,n=1,b=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 logarithmsLicenses & AttributionsCC licensed content, OriginalRevision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.CC licensed content, Shared previouslyPrecalculus. 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/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175..College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@5.2.