In this work, we consider computing the real logarithm of a real matri
x. We pay attention to general conditioning issues, provide careful im
plementation for several techniques including scaling issues, and fina
lly test and compare the techniques on a number of problems. All thing
s considered, our recommendation for a general purpose method goes to
the Schur decomposition approach with eigenvalue grouping, followed by
square roots and diagonal Pade approximants of the diagonal blocks. N
onetheless, in some cases, a well-implemented series expansion techniq
ue outperformed the other methods. We have also analyzed and implement
ed a novel method to estimate the Frechet derivative of the log, which
proved very successful for condition estimation.