By

*Euler's identity*,

, 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

for

complex 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

).

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