The log is the complex log, which is multiple-valued: , where n is any integer.
Is that what you were asking?
my problem, in its completeness, is the following: i'm looking for singularities of a function whose denominator is . So i look for solutions of . I put where is the principal branch of complex logarithm and by periodicity of complex exp i get , where log in RHS is the real log. So and i was looking for an easy way to write the RHS in order to compute residues at those points