What do you calculate logarithms manually without using a calculator?

Answer 1

for |x| <= 1 and x != -1, you can use a taylor series expansion:

ln(1 + x) = x * (1 - x * (1/2 - x*(1/3 - x *(1/4 - x * (1/5 - x* ...)))))

Unfortunately, this is only good for a narrow range of input values, and the rate of convergence is poor.

All other methods that I was able to find would essentially require the use of a calculator.