TITLE: log of summation expression
QUESTION [11 upvotes]: I am curious about simplifying the following expression:
$$\log \left(\sum_\limits{i=0}^{n}x_i \right)$$
Is there any rule to simplify a summation inside the log?

REPLY [3 votes]: There's this:
$$\log\left(\sum_{i=0}^n x_i\right) = \log(x_0) + \log\left(1+\sum_{i=1}^n\left(\frac{x_i}{x_0}\right)\right)$$
$x_0$ must be the biggest in the series.  It didn't help me with what I'm trying to solve, but I think it answers the question.