I'm interested in exponentiation of real numbers when the result is real. If exponentiation $x^y$ of real numbers $x$ and $y$ is defined as $e^{y \log x}$, then one cannot define things like $(-2)^4$.
How can one rigorously define such operations (without using the complex logarithm)?
Short answer:
When $y$ is rational, you use the "root" definition,
$$z=x^{p/q}\iff z^q=x^p$$
where $p/q$ is irreductible (no need to recall what a natural power is). For even $q$, $x$ and $z$ must be positive.
We need to add
$$x^{-p}=\frac1{x^p},\\x^0=1.$$
When $y$ is irrational, you don't define anything.
The reson why we don't define the powers of negative for irrational exponents is that we don't know what sign to choose to make it coherent with the ususal propertie of exponentiation. It doesn't seem possible to assign a "parity" to the reals.
If you want complex exponentiation then define $a^b = \exp(b\ln_π(a))$ where $\exp$ is the complex exponential function and $\ln_π$ is the principal branch of the logarithm, and it may be preferable to include the negative real line in the domain (but still excluding zero) in order that $\ln_π(-1) = iπ$ and $(-1)^{1/2} = \exp(iπ/2) = i$. Note that this still leaves $0^0$ undefined.
But if you want real exponentiation then firstly you probably want $\sqrt{0} = 0$ and secondly you also want $\sqrt[3]{-8} = -2$. This is incompatible with complex exponentiation. The most intuitive definition that satisfies these properties is as described here. It is possible to use the formula $a^b = \exp(b\ln(a))$ for reals $a,b$ such that $a>0$ and then patching in special cases, like $0^r = 0$ for positive rational $r$, and $(-a)^{p/q} = (-a^{1/q})^p$ for real $a > 0$ and integers $p,q$ where $q$ is odd and $\gcd(p,q) = 1$, but this is extremely ad-hoc and does not give any insight into why we want to do that, unlike the other way.
Finally, half-related to this issue is that we can define integer exponents on arbitrary division rings including complex numbers. This means that $(-2)^4 = 16$ in any division ring, where the "$2$" and "$16$" denote the appropriate sums of the ring's multiplicative identity, while the "$4$" is just a natural number.
It hss been mentioned here a number of times to define this in a weird way with extending casewise to certain rationals (i.e. $(-x)^{p/q}$ with odd $q$) and leaving irrationals undefined. I would argue though that this is not really the best definition in this case, and the best is to simply take
$$(-x)^y = (-x) \cdot (-x) \cdot \cdots \cdot (-x)\ (y\ \mathrm{copies\ of\ } (-x))$$
for $x < 0$ and $y \in \mathbb{N}$, extended to $y \in \mathbb{Z}$ by the usual recurrence, and undefined everywhere else. That is, define it only at integer $y$ and leave it undefined elsewhere entirely. (For that matter, leave the cube root $x^{1/3}$ undefined for $x < 0$, too, and all $x^{1/q}$ for odd $q$ and $x < 0$. We would say that $x^3 = -8$ has a solution $x = -2$, but this is not $(-8)^{1/3}$, rather we should more honestly write $x = -(8^{1/3})$ and note that it is a property of equations of the form $x^n = -a$ for $n$ odd and $a > 0$ that $x = -(a^{1/n})$ is a solution.)
I argue this is more natural for two reasons: One, if you use the other method, while you get more "exponents" you can use, you get a horribly ill-behaved function that doesn't resemble exponentials to positive bases. Undefined at densely many points, and if you plot it it looks like two "curves" that deceptively aren't really but something actually rather really freaky! Exponentials are nice functions -- why should they suddenly become awful for negative bases? Usually we don't encounter something so pathological in such an elementary context. Most often one associates similar pathology with the Dirichlet function
$$1_{\mathbb{Q}}(x) = \begin{cases}1\ \mbox{if $x$ is irrational} \\0\ \mbox{otherwise}\end{cases}$$
not usually mentioned until calculus!
For another, the truly best codomain for the object $(-x)^y$ is really the complex plane, and in that codomain it is truly a multi-valued "function" (better to call it an "association" or a "relation")
$$(-x)^y = e^{y(\log(x) + (2k + 1)\pi i)},\ k \in \mathbb{Z}$$
This is only unambiguously defined when $y$ is an integer. The "weird awful hideous thing" mentioned earlier actually would correspond to trying to take a section through this as a multivalued relation and not a function (although not quite -- there are two values for every even denominator and we still only picked one, but nonetheless "very many branches of this"). What "section" here means is that the above when graphed with respect to $y$ and in 3 dimensions with the other two being real and imaginary parts, forms a "spiral helix", actually infinitely many spiral helices covering the surface of a curved horn-like object kind of like a bunch of yarn wrapped around a spool. Where these helices of this whole multivalued thing intersect the plane formed by the "y"-axis and the real-part axis of the value, are essentially the highly pathological object mentioned before. To hit each "good" rational you have to use a different $k$ value, thus are looking at a section through a different helix. So in a sense choosing this highly pathological definition is a way to "sneak in a non-function disguised as a function" which doesn't seem right. We are also essentially forcing the complex numbers "in the back door" because these can only arise from complex exponentiations of $e$.
To echo Hadamard, the "nicest" path between two points of this thing, i.e. two integer points, passes through the complex domain, and there are many such nice paths, and if we are thus to stick to the real domain we should be honest and say there is no path at all, not to end up with a desperate and ultimately incomplete attempt to fill in that gap between the integers that amounts to trying to take all those paths at once.
EDIT: You ask how to make it consistent with $e^{y \log x}$ in the comments. This is what happens. If we take $(-x)^y = e^{y \log(-x)}$ then $\log(-x) = \log(x) + (2k+1)\pi i$ as before. The trick is now that $e^{(2k+1) \pi i} = -1$, so when you raise to an integer power $y$ it just becomes good ole' $(-1)^n$ (with $n = y$) in front of $e^{y \log(x)}$ which is real with a real valued log. There isn't any other way to make it consistent without going to complex numbers even for integer $y$ -- $\log(x)$ is not defined for $x < 0$ on the reals. This is another argument for limiting the definition.
One might even be tempted to go further and say we should therefore perhaps not even define $(-x)^y$ at all! Yet nonetheless, things like $(-1)^n$ for integer $n$ are very useful, e.g. in doing Taylor series expansions and combinatorial sums.
However that could still suggest another view, and that is that there is a distinction between "combinatorial" exponentiation -- which is the "repeated multiplication" definition as above, and "analytic" exponentiation, which is what "exponential functions to the base $b$", i.e. $b^x$, are, and these are defined by $e^{x \log(b)}$ and when $b$ is negative the combinatorial exponential exists but the analytic exponential doesn't, and these are two different animals. (And in fact that is what most textbooks which define an exponential function do: they explicitly state $b$ must be positive!) Indeed, you might also be interested in Qiaochu Yuan's post here:
https://math.stackexchange.com/a/56710/11172
This theme of being able to make distinctions between conceptually different but related things is I think something quite important. It also occurs in other contexts. For example, with physics, where I have argued that a long standing source of confusion -- "what is entropy? Disorder? Energy dispersal? Something else?" -- is best thought of in this way: that there are at least two conceptually very different things that "entropy" means -- one is directly physics related and it has to do with energy. Another meaning is from computer science and telecommunications science, information theory. It is then (and should be!) a surprising result that the two are in fact related on a very deep level, the elucidation of which represents an important part of statistical mechanics. But relation does not mean identity and the lumping together of conceptually distinct things and then insisting they are "the same" is something that I think very often confuses.
