log of summation expression

Sometimes we have asymptotics with most significant term $M \to \infty$ we do this $$ \log(M+A) = \log(M\cdot(1+S)) $$ so that $S=A/M$ is "small" in the sense $S=o(1)$, and then $$ \log(M\cdot(1+S)) = \log M + \log(1+S) = \log M + S - \frac{1}{2}S^2+\frac{1}{3}S^3+\cdots $$

reference
G. A. Edgar, Transseries for beginners, Real Anal. Exchange 35 (2010), no. 2, 253--309.

You may note that this is equivalent to trying to solve $$\log(a+b+c)$$You realize you can't do much about $a,b,$ or $c$.

The only way this can be simplified, is if you can factor something out and then apply log properties.

A more interesting question might concern $$\log\left(\Pi_{i=0}^na_i\right)=\sum_{i=0}^n\log(a_i)$$

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.