## 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
$\frac{1}{log_u m}+ \frac{1}{log_u n}= \frac{log_um+log_un}{log_um log_un}= \frac{log_u (mn)}{log_um log_un}$
.