log & exponent rules to solve attached equation

Question

I am having some trouble working out the maths on how to get from the first equation to the second equation in the attached photo where $\sigma^2=1$. I have absolutely no idea where to start, it looks so nasty!

Answer 1

We can use the rules:

$\log(ab) = \log(a) + \log(b)$, $\log(c^d) =d \log(c)$ and $\log(\exp f) = f$

So ignore the sum and fix an index $i$ for a moment, use the first rule with $a=\frac{1}{(2\pi)^{1/2}}$. Then use the second rule with $d = -1/2$. This gives you the first term in the answer. You can use the third rule (above) for the second term. Then sum over the indices $i$ - hope this helps!