Log(exp(z))=ln(|exp(z)|)+iarg(exp(z))=x+iarg(exp(z ))

z=x+iy

Now, e^z=exp(x)(cos(y)+isin(y))

So arg(e^z)=y iff y is between pi and minus pi or between 0 and 2pi, as long as it doesn't passes a length of 2pi, because then an increment of 2pi is added to the argument.