What Is Exponentiation?

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 $0<z<\infty$. 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.

$2^\pi$ or $3^\sqrt2$ (or any other irrational power, really). What does this mean?

$$a^\pi=a^{3.1415\ldots}=a^{3\ +\ 0.1\ +\ 0.04\ +\ 0.001\ +\ 0.0005\ +\ \cdots}=a^3\cdot a^{0.1}\cdot a^{0.04}\cdot a^{0.001}\cdot a^{0.0005}\cdots$$

$$a^\sqrt2=a^{1.4142\ldots}=a^{1\ +\ 0.4\ +\ 0.01\ +\ 0.004\ +\ 0.0002\ +\ \cdots}=a^1\cdot a^{0.4}\cdot a^{0.01}\cdot a^{0.004}\cdot a^{0.0002}\cdots$$

It is obvious that the general factor of this infinite product tends towards $a^0=1$. Convergence then follows from the fact that each single decimal digit is in between $0$ and $9$, meaning that $\displaystyle\prod_ka^{^{\tfrac{d_k}{10^k}}}$ is in between $\displaystyle\prod_ka^{^{\tfrac0{10^k}}}=1$, and $\displaystyle\prod_ka^{^{\tfrac9{10^k}}}=a^{10^{-n}}$, where $n$ is the number of digits of $[a]$.

One reason your intuition might break down here is that exponentiation - even integer exponentiation - is inherently dimensionless. If a square has area of $5\mathrm{cm}^2$ then you can talk about its sides having length $\sqrt{5\mathrm{cm}^2} = \sqrt{5}\mathrm{cm}$, but there are no units for $x$ that give $2^x$ a sensible set of units — try to figure out the units in the formula $2^{a+b} = 2^a2^b$. That said, this doesn't mean that there can't be understanding; there are many different ways of understanding something. But the physical understanding is going to be difficult.

OTOH, 'difficult' isn't necessarily 'impossible'; one reasonable approach to understanding the exponential is via the classic rate-change model. If I have $1$kg of paramecia and they're doubling their mass every minute, what will be the mass of the paramecia after $2.35$ minutes? Note that this gets around the dimension barrier I mentioned in the previous paragraph, because the quantity being exponentiated is dimensionless; it has dimensions of (time)/(time) (with the numerator being the elapsed time and the denominator being the doubling time).

With that said, though, this approach is going to be effectively impossible to extend to complex exponents (what would $5+2i$ time intervals be?), and my hearty recommendation would be to try and get away from the physical understanding in your head; don't map the exponential to something that you have 'real-world' experience with, but instead try to understand it on its own terms and understand its properties directly.