Contrary to popular belief, you can do better that power series. The trick is in the use of continued fractions and the related Padé approximants.
One continued fraction for the logarithm (due to Khovanskiĭ) goes a bit like this:
The beauty of this is that it has a wider domain of applicability: it is valid as long as $|\arg(1+z)| < \pi$.
One can use the Lentz-Thompson-Barnett method on this CF, of course, but one could also choose to exploit argument reduction here, by suitably exploiting the identity $\log(ab)=\log\,a+\log\,b$. If you take that route, you can be justified in just using a truncation of the continued fraction. That truncation is what's called a Padé approximant.
I'll edit later with more details if needed.