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  
\[log_a b = \frac{log_x b }{log_x a }\]
\[log_a b = \frac{log_b b }{log_b a } = \frac{1}{log_ba}\]
Suppose the that we want to simplify  
\[log_m u+ log_nu\]
Using the change of base rule as above gives  
\[\frac{1}{log_u m}+ \frac{1}{log_u n}= \frac{log_um+log_un}{log_um log_un}= \frac{log_u (mn)}{log_um log_un}\]

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