By Euler’s formula, , so that
from which it follows that for any , .
Similarly, , so that
and for any imaginary number , , where is real.
Finally, from the polar representation forcomplex numbers,
where and 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 extracts its phase (times ).
Another usage is in Homomorphic Signal Processing [8, Chapter 10] in which the multiplicative formants in vocal spectra are converted to additive low-frequency variations in the spectrum (with the harmonics being the high-frequency variation in the spectrum). Thus, the lowpass-filtered log spectrum contains only the formants, and the complementarily highpass-filtered log spectrum contains only the fine structure associated with the pitch.Exercise: Work out the definition of logarithms using a complex base .