With double precision, I get $\log(\exp(-3)+1)=0.048587351573741958$, which already has $4$ incorrect digits, and $\log(\exp(-30)+1)=9.348... \times10^{-14}$, which only has two correct digits.
What is the most elegant way to circumvent this? I tried a Taylor series of $\log(y)$ around $y=1$, but even with 14 terms I still didn't get all digits correctly for $x=-3$.
Use the (IEEE standard) library function log1p, which should be present in all programming languages. The function log1p(x) returns $\log(1+x)$, and is implemented with particular attention to accuracy when $x$ is small. It is designed to solve exactly this kind of problem.
If you want to use your own implementation to evaluate the function, take a look on this article. Otherwise, just go with log1p as suggested by @Frederico Poloni in his answer.
For more information about the log1p function read this other article from the same author.
