The matrix equation log(XY)=log(X)+log(Y)

Let X,Y be n×n complex matrices such that X,Y,XY have no eigenvalues on R- and log(XY)=log(X)+log(Y). We prove that if n=2, or if n⩾3 and X,Y are simultaneously triangularizable, then X,Y commute. In both cases we reduce the problem to a result in complex analysis.