What Is Exponentiation?

Is there an intuitive definition of exponentiation?

In elementary school, we learned that

where $b$ is an integer.

Then later on this was expanded to include rational exponents, so that

From there we could evaluate decimal exponents like $4^{3.24}$ by first converting to a fraction.

However, even after learning Euler’s Identity, I feel as though there is no discussion on what exponentiation really means. The definitions I found are either overly simplistic or unhelpfully complex. Once we stray from the land of rational powers into real powers in general, is there an intuitive definition or explanation of exponentiation?

I am thinking along the lines of, for example, $2^\pi$ or $3^{\sqrt2}$ (or any other irrational power, really). What does this mean? Or, is there no real-world relationship?

To draw a parallel to multiplication:
If we consider the expression $e\cdot \sqrt5$, I could tell you that this represents the area of a rectangle with side lengths $e$ cm and $\sqrt5$ cm. Or maybe $e \cdot \pi$ is the cost of $\pi$ kg of material that costs $e$ dollars per kg.
Of course these quantities would not be exact, but the underlying intuition does not break down. The idea of repeated addition still holds, just that fractional parts of terms, rather than the entire number, are being added.

So does such an intuition for exponentiation exist? Or is this one of the many things we must accept with proof but not understanding?

This question stems from trying to understand complex exponents including Euler’s identity and $2^i$, but I realized that we must first understand reals before moving on the complex numbers.

My chief understanding of the exponential and the logarithm come from Spivak’s wonderful book Calculus. He devotes a chapter to the definitions of both.

Think of exponentiation as some abstract operation $f_a$ ($a$ is just some index, but you’ll see why it’s there) that takes a natural number $n$ and spits out a new number $f_a(n)$. You should think of $f_a(n) = a^n$.

To match our usual notion of exponentiation, we want it to satisfy a few rules, most importantly $f_a(n+m) = f_a(n)f_a(m)$. Like how $a^{n+m} = a^na^m$.

Now, we can extend this operation to the negative integers using this rule: take $f_a(-n)$ to be $1/f_a(n)$. then $f_a(0) = f_a(n-n) = f_a(n)f_a(-n) = 1$, like how $a^0=1$.

Then we can extend the operation to the rational numbers, by taking $f_a(n/m) = \sqrt[m]{f_a(n)}$. Like how $a^{n/m} = \sqrt[m]{a^n}$.

Now, from here we can look to extend $f_a$ to the real numbers. This takes more work than what’s happened up to now. The idea is that we want $f_a$ to satisfy the basic property of exponentiation: $f_a(x+y)=f_a(x)f_a(y)$. This way we know it agrees with usual exponentiation for natural numbers, integers, and rational numbers. But there are a million ways to extend $f_a$ while preserving this property, so how do we choose?

Answer: Require $f_a$ to be continuous.

This way, we also have a way to evaluate $f_a(x)$ for any real number $x$: take a sequence of rational numbers $x_n$ converging to $x$, then $f_a(x)$ is $\lim_{n\to\infty} f_a(x_n)$. This seems like a pretty reasonable property to require!

Now, actually constructing a function that does this is hard. It turns out it’s easier to define its inverse function, the logarithm $\log(z)$, which is the area under the curve $y=1/x$ from $1$ to $z$ for $0. Once you've defined the logarithm, you can define its inverse $\exp(z) = e^z$. You can then prove that it has all the properties of the exponential that we wanted, namely continuity and $\exp(x+y)=\exp(x)\exp(y)$. From here you can change the base of the exponential: $a^x = (e^{\log a})^x = e^{x\log a}$.

To conclude: the real exponential function $\exp$ is defined (in fact uniquely) to be a continuous function $\mathbb{R}\to\mathbb{R}$ satisfying the identity $\exp(x+y)=\exp(x)\exp(y)$ for all real $x$ and $y$. One way to interpret it for real numbers is as a limit of exponentiating by rational approximations. Its inverse, the logarithm, can similarly be justified.

Finally, de Moivre's formula $e^{ix} = \cos(x)+i\sin(x)$ is what happens when you take the Taylor series expansion of $e^x$ and formally use it as its definition in the complex plane. This is more removed from intuition; it's really a bit of formal mathematical symbol-pushing.