Principal Logarithm

Definition

The principal logarithm is the function $\mathrm{Log}:\mathbb{C}\setminus \{0\}\to \mathbb{C}$ is defined by
$\mathrm{Log}z=\ln |z|+i\mathrm{Arg}z.$

For $z=x>0$

We get $\mathrm{Log}x=\ln x$.