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

).
Next Section: Properties of DB ScalesPrevious Section: Changing the Base