Q:

Why is exponentiation defined as $x^y=e^{\ln(x)\cdot y}$?

There are many curves that extend integer exponentiation to larger domains, so why was this one chosen?

A:

It's economic.  You define exponentiation to one base, say $e$,  rigorously and reduce all other exponentiation to that case.
Edit: The reason why mathematicians chose just $e$ instead of $42$ for that purpose has reasons which go way beyond the scope of the OP's question.

A:

I believe there is only one function $f(x, y)$, $f: \mathbb{R}^{+} \times \mathbb{R} \rightarrow \mathbb{R}$ satisfying

$f$ is continuous,
$f(x, 1) = x$,
$f(x, y + z) = f(x, y) f(x, z)$,

and that is $f(x, y) = x^y = e^{\ln(x) y}$, where $e^x$ is defined by the exponential function. Note that 2) and 3) fix the values of the function at rational $y$ (e.g. you can see that $f(x, 1/3) = \sqrt[3]{x}$, for example) with odd denominator in lowest terms, and 1) fixes them at the other rational and at irrational $y$. 3) is just the exponent law $x^{n + m} = x^n x^m$ for integer exponents assumed to hold for real ones.

A:

Because it has some very useful properties. For example
$$ \frac{d}{dx} e^x = e^x $$
but with another base you would need a constant.
An analogy is using radians instead of degrees once you get more advanced at trigonometry. Here again
$$ \frac{d}{dx} \sin x = \cos x $$
when $x$ is in radians. In other representations you need a constant.
Unlike the radians case, $e$ has some other very special properties that are both practical and worthy of study for their own sake.
For example $e^x$ has a very simple infinite series representation:
$$ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots $$
where $n!$ is factorial($n$).
and there is a relationship between $e$ and the trigonmetric functions $\sin$ and $\cos$:
$$ e^{ix} = \cos x + i\sin x $$
where $i$ is the imaginary number $\sqrt{-1}$.
So switching to using $e$ has some practical benefits at first, but as you go further it becomes quite fundamental to many branches of mathematics.

