Calculate logarithms by hand

I'm thinking of making a table of logarithms ranging from 100-999 with 5 significant digits. By pen and paper that is. I'm doing this old school.

What first came to mind was to use $\log(ab) = \log(a) + \log(b)$ for reduction.
And then use the taylor series for $\log(1-x)$ when $-1 < x \leq 1$ But convergence is rather slow on this one.

Can you come up with a better method?

For $1 \le x \le 2$, $$\begin{eqnarray*} \ln(x) &\approx - 1.941064448+ \left( 3.529305040+ \left( - 2.461222169+ \left( \right.\right.\right.\cr & \left.\left.\left. 1.130626210+ \left( - 0.2887399591+ 0.03110401824\,x \right) x \right) x \right) x \right) x \end{eqnarray*}$$ with error less than $10^{-5}$. For $2^n \le x \le 2^{n+1}$, $\ln(x) = n \ln(2) + \ln(x/2^n)$.

According to Wikipedia, http://en.wikipedia.org/wiki/Logarithm#Power_series, you can try $$\ln(z)=2\sum_{n=0}^\infty\,\frac{1}{2n+1}\left(\frac{z-1}{z+1}\right)^{2n+1}$$ And using that convergence is quickler for $z$ near to $1$, according to wikipedia for $z=1.5$ the first three terms of the series give an error of about $3\cdot 10^{-6}$.