For \(z\in\CC\setminus\{0\}\), let \(\Arg(z)\in(-\pi,\pi]\) be the principal argument of \(z\), the unique \(\theta\in(-\pi,\pi]\) with \(z=\abs{z}e^{i\theta}\). The principal value of the logarithm is \[ \log(z)\coloneqq\ln\abs{z}+i\Arg(z),\qquad z\in\CC\setminus\{0\}, \] where \(\ln\) is the real natural logarithm.
In polar form \(z=re^{i\theta}\) with \(r>0\) and \(\theta\in(-\pi,\pi]\), this reads \(\log(z)=\ln r+i\theta\). It satisfies \(e^{\log(z)}=z\), restricts to the principal branch \(\Log\) on \(\CC\setminus(-\infty,0]\), and is discontinuous at every point of \((-\infty,0)\), where \(\Arg\) jumps from values near \(-\pi\) to \(\pi\).