The Complex Logarithm Is Multivalued

Inverting a function that repeats

To take a logarithm is to undo an exponential: solving log z means finding every w with e^w = z. On the real line this is harmless, because e^x is strictly increasing — each positive output comes from exactly one input, so ln x is a single number. But you just met a startling fact in the previous guides: the complex exponential is periodic, with e^(w + 2 pi i) = e^w for every w. A function that repeats cannot have a single-valued inverse, and that one observation is the whole story of this guide.

Suppose w is one solution of e^w = z. Then so is w + 2 pi i, and w + 4 pi i, and w - 2 pi i, and so on forever. The logarithm is multivalued: it does not hand you one answer but an infinite ladder of them, each rung exactly 2 pi i above the last. They share the same real part and differ only in their imaginary parts, marching up and down a vertical line in the w-plane. So writing "log z" without further care names not a value but a whole set.

Reading off the formula from polar form

The cleanest way to pin down every solution is to write z in polar form. If z = r e^(i theta) with r = |z| and theta an argument of z, and we seek w = u + i v with e^w = z, then e^u e^(i v) = r e^(i theta). Matching moduli gives e^u = r, so u = ln r — an ordinary real logarithm of a positive number. Matching directions gives v = theta, but only up to whole turns, since rotating by 2 pi lands you back where you started. That is precisely where the ladder of 2 pi i comes from.

z = r e^(i theta),   r = |z| > 0

log z = ln r + i (theta + 2 pi k),   k = 0, +-1, +-2, ...
      = ln |z| + i arg z       (arg z is itself a whole set)

Two things are worth pausing on. First, the real part ln |z| is completely determined — all the multivaluedness lives in the imaginary part, which is just the argument of z with its built-in 2 pi ambiguity. Second, the formula demands r > 0, which is no accident: it is the planar echo of the fact that the exponential is never zero. Since e^w can never equal 0, the equation e^w = 0 has no solution, so log 0 is genuinely undefined. The origin is the one point in the plane the logarithm simply refuses to touch.

A worked example: the logarithm of -1

Real calculus tells beginners that you cannot take the logarithm of a negative number. The complex plane cheerfully disagrees — you can, you just get infinitely many answers. Take z = -1. Its modulus is |z| = 1, so ln |z| = ln 1 = 0, and its arguments are pi, pi + 2 pi, pi - 2 pi, and so on. Following the formula, log(-1) = i (pi + 2 pi k) for every integer k. The famous slogan that 'e raised to i pi equals -1' is just one rung of this ladder read backwards.

1. Write z in polar form: -1 = 1 * e^(i pi), so r = 1 and one argument is theta = pi.
2. Real part: u = ln r = ln 1 = 0 — every logarithm of -1 sits on the imaginary axis.
3. Imaginary part: v = theta + 2 pi k = pi + 2 pi k, sweeping through ..., -3 pi, -pi, pi, 3 pi, ....
4. Collect: log(-1) = i (pi + 2 pi k) = ..., -3 pi i, -pi i, pi i, 3 pi i, ... — infinitely many honest values.

Taming the ladder: the principal Log

An infinite set is awkward to compute with, so we usually pick one rung and call it the answer. The standard choice is the principal logarithm, written with a capital L as Log z. It uses the principal argument Arg z, the single value of arg z chosen to lie in the half-open interval (-pi, pi]. Then Log z = ln |z| + i Arg z, and on the positive real axis this agrees exactly with the ordinary ln you already know, so the new notion extends the old one rather than replacing it.

Choosing one value out of the set is exactly what it means to select a branch of the logarithm. The principal branch is the most common choice, but it is a choice, not a law of nature — for a problem living near the negative real axis you might prefer an argument range like [0, 2 pi) instead. What matters is to fix one range and stay loyal to it, because the next guide will show that quietly switching branches mid-calculation is one of the classic ways to produce a confident, completely wrong answer.

Why no branch can survive a full loop

Here is the deep reason the multivaluedness can never be fully swept away. Imagine walking the point z once counterclockwise around a small circle centred at the origin. Its modulus |z| returns to its starting value, so the real part ln |z| of the logarithm comes back unchanged. But its argument increases steadily and ends up 2 pi larger than it began — so the imaginary part of log z is 2 pi greater after the loop. You return to the same z but a different value of the logarithm. The origin is a branch point: a point you cannot encircle without the function jumping to the next rung.

This is why any single-valued choice — the principal Log included — must be torn somewhere. To keep the principal branch continuous you forbid z from ever crossing the negative real axis, where Arg z leaps from just below pi to just above -pi. That forbidden ray is called a branch cut, the topic the very next guide opens up in full. The honest summary: there is no continuous single-valued logarithm on the whole punctured plane. You can have continuity, or you can have a single value everywhere around the origin, but not both at once.

There is a beautiful way to keep everything continuous AND single-valued at once: stop insisting the answer lives in a flat plane. If you let z wind around the origin and spiral upward onto a new sheet each time, you build the Riemann surface of the logarithm — an endless helical ramp on which log becomes a genuine, perfectly continuous function. On that surface the ladder of values is not a defect but the natural geometry. We only meet trouble when we flatten the ramp back onto one plane and are forced to cut it.