Definition of $x^n$ where $x\leq 0$

Question

This is a follow up to this MathEducators question. I'm trying to define exponents through the route recommended by the leading answer:

1. Define $\ln(x) = \int_1^x \frac{1}{t} dt$.
2. Define $\exp$ as the inverse of $\ln$.
3. Define $x^n = \exp(n\ln(x))$.

This works well for $x>0$, but I'm having trouble seeing how this would work for $x\leq0$. As defined, $\ln (x)$ is undefined for $x\leq0$.

Am I missing something? Or is there something we must do to patch up the above series of definitions, so that our definition of $x^n$ allows for $x\leq0$?

Answer 1

For positive numbers $x$, there is no problem at all to define $x^{a/b}$ for a rational number $a/b$. It is $\sqrt[b]{x^a}$ or $(\sqrt[b]{x})^a$. With this definition, all the usual laws of exponents are extended to rational exponents. To go beyond this, for real exponents $a$, one can define $x^a = \exp(a \ln(x))$, as you have done.

For negative $x$ there is no problem with integer exponents, but rational exponents are already subtle and involve complex numbers as you can see by the example $(-1)^{1/2}$, which has to be the imaginary number $i$ or $-i$. For the general solution one has to extend $\ln$ and $\exp$ to the complex domain, and this involves some further subtleties.

Answer 2

You'll need complex numbers for this, and because of this you'll have to live with branch cuts.

$\ln(z)$ for negative real $z$ can be considered $\ln(-z)+\pi i$. Which does crazy things: you only get a real value for integer $n$, because those are the only ones where you're guaranteed to get an integer out.

Buuut it's not just $\pi i$ really: it's $(2k+1)\pi i$ for $k \in \mathbb Z$, because all of them work approximately the same way. Similarly, for any complex numbers, $\ln(z) = \ln(|z|) + (\arg(z) + 2k\pi)i$, which includes for positive reals. This has some interesting results: exponentiation to a non-integer gives a multitude of possible outputs; for rational exponents there are as many results as the denominator of the fraction, and for irrational exponents there are infinitely many.

For the positive reals, because we can think in terms of real numbers anyway, the others can be (mostly) safely ignored, but once we get into complex numbers and the negative reals, because $\ln$ works this way, we are stuck with many possible answers.