TITLE: Product of logarithms, prove this identity.
QUESTION [7 upvotes]: Is it hard to prove this identity:
$$2 \log (a) \log (b)=\log(a b)^2-\log(a)^2-\log(b)^2$$
for $a>1$ and $b>1$?

REPLY [20 votes]: $$\begin{align}(\log ab)^2 &= (\log a + \log b)^2 \\
&= (\log a)^2 + (\log b)^2 + 2\log a \log b \end{align}$$
Now, just rearrange by leaving $2\log a \log b$ on the right hand side.