numerical methods - 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?