## Multiplication Rule for Logarithm Bases

Is there a logarithm rule that allows us to multiply the bases of logarithms?Yes there is.

To derive it use the change of base rule

\[log_a b = \frac{log_x b }{log_x a }\]

.If

\[x=b\]

then \[log_a b = \frac{log_b b }{log_b a } = \frac{1}{log_ba}\]

.Suppose the that we want to simplify

\[log_m u+ log_nu\]

.Using the change of base rule as above gives

\[\frac{1}{log_u m}+ \frac{1}{log_u n}= \frac{log_um+log_un}{log_um log_un}= \frac{log_u (mn)}{log_um log_un}\]

.