How do we come to know what to add in the below question on logarithm?

Question

If $a^2 + b^2 = 3ab$ , show that $\log(\frac{a+b}{\sqrt 5})= \frac{1}{2}(\log a + \log b) + 2 \log 2$

Here, the first step is to add $2ab$ on both sides and then continue solving. How will one come to know if $2ab$ needs to be added or $7ab$ needs to be added or for that matter something else needs to be added?

Answer 1

We have $$(a+b)^2=2ab+3ab\iff\left(\dfrac{a+b}{\sqrt5}\right)^2=ab$$

Now apply logarithm and use $\log(pq)=\log p+\log q$ and $\log x^m=m\log x$

I fear there is some mistake in the proposition.