complex logarithm
The real logarithm undoes the real exponential, one clean answer for each positive input. The complex logarithm tries to undo the complex exponential — but that exponential is periodic, wrapping the plane around and around, so its inverse cannot have a single value. The complex logarithm is therefore inherently multivalued: every nonzero number has infinitely many logarithms, differing by multiples of 2 pi i, like the floors of a spiral staircase stacked above one ground-floor answer.
Writing z in polar form as z = r e^{i theta}, the logarithm is log z = ln r + i theta, where ln r is the ordinary real logarithm of the modulus and theta is the argument (angle) of z. The catch is that theta is only defined up to adding any multiple of 2 pi, since rotating by a full turn lands on the same point. To get a single-valued function you choose a principal value, usually by restricting the angle to the interval (-pi, pi], and writing Log z; this forces a branch cut, conventionally along the negative real axis, across which the function jumps by 2 pi i. The origin, where r = 0, is a branch point and must be excluded.
The complex logarithm is the source of complex powers (since z^a is defined as e^{a log z}) and the reason fractional and irrational powers are multivalued. It appears whenever an integrand has a 1/z factor: integrating around a loop enclosing the origin adds 2 pi i, the residue mechanism in disguise. In applied work it governs phase unwrapping in signals, the impedance of circuits, and the potential of a line charge or vortex — anywhere a quantity accumulates an angle as you circle a singularity.
Take z = -1, which has modulus 1 and angle pi. Then log(-1) = ln 1 + i(pi + 2 pi k) = i pi (1 + 2k) for every integer k: ..., -i pi, i pi, 3 i pi, .... The principal value is Log(-1) = i pi. So a real negative number does have a complex logarithm — there is no contradiction, only multivaluedness.
Every nonzero number has infinitely many logarithms, spaced 2 pi i apart; the principal value picks one floor.
Familiar identities like log(z w) = log z + log w hold only up to multiples of 2 pi i, and can fail outright for the principal value Log. Blindly applying real-variable log rules in the complex plane is a common and costly mistake.