<s>[INST] If $P=log_{x}xy $ and $Q=log_{y}xy$, then how is $P+Q=PQ$? If $P$ and $Q$ have different bases for the log, how do we prove $P+Q=PQ$?
 [/INST] Consider $P + Q$ , it is $log_{x}{xy}+log_{y}{xy}$ which is equal to
$ log_{x}{x} + log_{x}{y} + log_{y}{x} + log_{y}{y}= 2 + log_{x}{y} + log_{y}{x} $
now consider rightside $P.Q$ which is $log_{x}{xy} .log_{y}{xy} = (1 + log_{x}{y}).(1+log_{y}{x})  $
which is equal to $1 + log_{x}{y} + log_{y}{x} + log_{x}{y}.log_{y}{x}$  and this last term will be $1$ , equal to $2+log_{x}{y}+log_{y}{x}$,
Comparing bothsides you see they are equal.Hence, proved.  
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