In this paper we will be interested in characterizing and computing matrices $X;{in};C^{n{ imes}n}$ that satisfy $e^X$ = A, that is logarithms of A. The study in this work goes through two lines. The first is concerned with a theoretical study of the solution set, S(A), of $e^X$ = A. Along the second line computational approaches are considered to compute the principal logarithm of A, LogA.