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?
Best Answer
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 $2^n \le x \le 2^{n+1}$, $\ln(x) = n \ln(2) + \ln(x/2^n)$.