Product of logarithms, prove this identity.

Question

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 $b>1$?

Judging by Yiyuan Lee and Umberto P.'s answer,the correct answer should be "no". — rah4927, Mar 21 '14 at 17:39

score 20 · Accepted Answer · answered Mar 21 '14 at 13:35

20

$$\begin{align}(\log ab)^2 &= (\log a + \log b)^2 \\ &= (\log a)^2 + (\log b)^2 + 2\log a \log b \end{align}$$

Now, just rearrange by leaving $2\log a \log b$ on the right hand side.

answered Mar 21 '14 at 13:35

Yiyuan Lee

score 15 · Answer 2 · answered Mar 21 '14 at 13:19

15

Here's a hint: $A^2 + 2AB +B^2 = (A+B)^2$. Here's another hint: $\log a + \log b = \log(ab)$.

answered Mar 21 '14 at 13:19

Umberto P.

2 Answers2