Logarithms in Summations : Confusion!

Question

I see this simplification and I am confused! I thought there is no explicit way to simplify the logarithm of a summation.

Can someone explain how the the second term( involving the summation), gets converted to a log of a summation of exponential log sums ?

The equation is in the image below! Thanks! :)

enter image description here

Answer 1

This is just an application of the fact that $\log(a)+log(b) = \log(ab)$.

Answer 2

The main trick is as follows:

\begin{align} a_ib_i &= \exp\left(\log(a_ib_i)\right) \\ &= \exp \left( \log(a_i) + \log(b_i) \right) \end{align}

if $a_i, b_i >0$.