$\sum\limits_{n=1}^\infty \log(1+a_n)$ converges absolutely $\iff\sum\limits_{n=1}^\infty a_n$ converges absolutely.

Hint: From the definition of $\ln'(1),$ we have

$$\lim_{u\to 0}\frac{\ln (1+u)}{u} = 1.$$

Thus there is $a>0$ such that

$$\frac{1}{2}\le \left|\frac{\ln (1+u)}{u}\right| \le \frac{3}{2}$$

for $u\in (-a,a),u\ne0.$