This is motivated by an earlier question of mine, in which I realized I was never really presented a definition of $e^x$, or more generally, what it means to raise a (positive) real number to an irrational power.

I know that the definition of $a^b$ with $a \in \mathbb{R}^+, b \in \mathbb{Q}$ is pretty straightforward in terms of repeated multiplication and the property that $a^{bc}=(a^b)^c$. I also know that one can define $a^b$ where $b \in \mathbb{R} - \mathbb{Q}$ using limits. This is stated, for example, in this Math.SE question.

Other way to define exponentiation with real powers is with the function $\exp(x)$ or $e^x$, which has many equivalent definitions. For example, one may define it as $e^x = \lim\limits_{n \to \infty} (1+\frac{x}{n})^n$, or as the unique solution to $y' = y$ with $y(0)=1$. Wikipedia has a whole page stating these definitions and showing that they are equivalent to each other.

What I haven't seen is a proof that this new $e^x$ behaves just like the old way of doing exponentiation when $x \in \mathbb{Q}$. If I were to guess, I'd say it's related to Wikipedia's fifth defintion: it is the unique (with some conditions) function that satisfies $f(1) = e$ and $f(x+y)=f(x)f(y)$. However, that defintion seems to involve more advanced concepts than the other ones, concepts which I don't really understand right now.

Is there a proof of the fact that $\exp(x)$ is equivalent to the definition of exponentiation for rational powers?