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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

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  • $\begingroup$ $e^{(\log{a}+\log{b})}=e^{\log{a}}e^{\log{b}}=ab$ $\endgroup$ – saulspatz Nov 26 '18 at 13:36
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This is just an application of the fact that $\log(a)+log(b) = \log(ab)$.

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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 $a_i, b_i >0$.

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