defined by branch cuts on the real axis from negative infinity to -1 and from 1 to infinity, f(0)=0
show by setting that f maps the disc |z|<1 onto the infinite strip
here for a longer discussion. Now in that thread it is show that maps the disk onto the right half-plane . But you asking about . This is not a problem if we do the necessary adjusments. We can think of as first and then followed by . However, is an automorphism of the disk and so maps the disk onto the half-plane . From here it ought to be clear that the log maps the half-plane to a infinite strip (that is what logs always do as conformal mappings).