Logarithms of Negative and Imaginary Numbers

By Euler's identity, $e^{j\pi} = -1$, so that

$$\ln(-1) = j\pi$$

from which it follows that for any $x<0$, $\ln(x) = j\pi + \ln(\vert x\vert)$.

Similarly, $e^{j\pi/2} = j$, so that

$$\ln(j) = j\frac{\pi}{2}$$

and for any imaginary number $z = jy$, $\ln(z) = j\pi/2 + \ln(y)$, where $y$ is real.

Finally, from the polar representation $z=r e^{j\theta}$ for complex numbers,

$$\ln(z) \isdef \ln(r e^{j\theta}) = \ln(r) + j\theta$$

where $r>0$ and $\theta$ are real. Thus, the log of the magnitude of a complex number behaves like the log of any positive real number, while the log of its phase term $e^{j\theta }$ extracts its phase (times $j$).