Multiplying two logarithms 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?
$$
\log(x) = \log(100x) \, \log(2)
$$

Fill in details after you check the basic properties of logarithms, and assuming $\;\log=\log_{10}\;$:
$$\log x=\log 100x\cdot\log2=\left(\log100+\log x\right)\log2\implies$$
$$(1-\log2)\log x=2\log2\implies\ldots$$
