# What is the quotient rule of logarithms?

Sep 10, 2014

The answer is $\log \left(\frac{a}{b}\right) = \log a - \log b$ or you can use $\ln \left(\frac{a}{b}\right) = \ln a - \ln b$.

An example of how to use this: simplify using quotient property: $\log \left(\frac{{2}^{5}}{{2}^{2}}\right)$

$= \log \left({2}^{5}\right) - \log \left({2}^{2}\right)$
$= 5 \log 2 - 2 \log 2$
$= 3 \log 2$

Or you could have a problem in reverse: express as a single log: $2 \log 4 - 3 \log 5$

$= \log \left({4}^{2}\right) - \log \left({3}^{5}\right)$
$= \log \left(16\right) - \log \left(125\right)$
$= \log \left(\frac{16}{125}\right)$