TITLE: Multiplying two logarithms
QUESTION [6 upvotes]: I've searched for some answer already, but couldn't find any solution to this problem. Apparently, there's no rule for the product of two logarithms. How would I then find the exact solution of this problem?
$$
\log(x) = \log(100x) \, \log(2)
$$

REPLY [7 votes]: How would I then find the exact solution of this problem?

Manipulate the equation to isolate $x$.
\begin{align*}
\log(x) &= (\log(100)+\log(x))\log(2)  \\
\log(x) &=\log(100)\log(2)+\log(x)\log(2)\\
\log(x)-\log(x)\log(2)&=\log(100)\log(2)\\
\log(x)(1-\log(2))&=\log(100)\log(2)  \\
\log(x)&=\log(100)\log(2)/(1-\log(2))\\
\end{align*}
Then resolve $x$ with whatever base your logarithm is using. E.g. with base 10, 
$$x\approx7.267$$