Complex Logarhytmic Function

The logarithmic function of a Complex Number $z=a+ib=r{e}^{i\theta }$ is $logz=logr{e}^{i\theta }=logr+log{e}^{i\theta }$. If we take the natural logarithm, this simplifies to $log\text{ }z=logr+i\theta$.

Principal Logarithm

The principal logarithm, denoted by $Logz$ is constrained by $-\pi <\theta \le \pi$ and gives only one principal value, whereas $logz=w$ gives all possible $w$ that satisfy $e^w=z$ (and differ from the principal solution by $2\pi n$).