No, using complex numbers alone DOES NOT work. To really allow complex numbers, you also need Riemann surfaces. The function "ln" has type ln: R -> CC where "R" denotes the Riemann surface corresponding to the natural domain of ln, and "CC" denotes the complex numbers. See here for details: https://en.wikipedia.org/wiki/Complex_logarithm#The_associat...

You can also allow it to be multi-valued and consider a principal branch when needed, the same we do when we discuss roots of monomials in algebra. The two situations are identical (as they must be, because logarithms generalize roots).