What algorithm is used by computers to calculate logarithms?

Solution 1:

It really depends on the CPU.

For intel IA64, apparently they use Taylor series combined with a table.

More info can be found here: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.5177

and here: http://www.computer.org/csdl/proceedings/arith/1999/0116/00/01160004.pdf

Solution 2:

All methods I have seen reduce first by dividing by the power of $2$ in the exponent and adding that exponent times $\ln(2)$ to the result. From there, the two most common methods I have seen are Taylor series for $\ln(1+x)$ and a variant of the CORDIC algorithm.

J. M. also brings up Padé approximants which I have seen used in some calculator implementations.