TITLE: Why do we need to prove $e^{u+v} = e^ue^v$?
QUESTION [30 upvotes]: In this book I'm using the author seems to feel a need to prove
$e^{u+v} = e^ue^v$
By
$\ln(e^{u+v}) = u + v = \ln(e^u) + \ln(e^v) = \ln(e^u e^v)$
Hence $e^{u+v} = e^u e^v$
But we know from basic algebra that $x^{a+b} = x^ax^b$.
Earlier in the chapter the author says that you should not assume $e^x$ "is an ordinary power of a base e with exponent x."
This is both a math and pedagogy question then, why does he do that?
So 2 questions really

Do we need to prove this for such a basic property?
If we don't need to, then why does he do it?  Fun?  To make it memorable?  Establish more neural connections?  A case of wildly uncontrolled OCD?

Also I've always taken for granted the property that $x^{a+b} = x^a x^b$.  I take it as an axiom, but I actually don't know where that axiom is listed.

REPLY [2 votes]: One definition of $e^x$ is $\displaystyle \lim_{n \rightarrow \infty}(1 + \frac{x}{n})^n$. From this definition, it doesn't automatically follow that $e^x e^y = e^{x+y}$.
In fact, it doesn't even follow immediately that $e^x = \displaystyle \lim_{n \rightarrow \infty}(1 + \frac{x}{n})^n = (\displaystyle \lim_{n \rightarrow \infty}(1 + \frac{1}{n})^n)^x = (e)^x$. What this means is $e^x$ is just a short hand notation for the limit which after some analysis we realize it as $(e)^x$.
By limit arguments, we can now show that $\displaystyle \lim_{n \rightarrow \infty}(1 + \frac{x}{n})^n = 1 + \sum_{k=1}^{\infty} \frac{x^k}{k!}$, $\forall x \in \mathbb{R}$.
Now $e^x \times e^y = (1 + \sum_{k=1}^{\infty} \frac{x^k}{k!}) \times (1 + \sum_{k=1}^{\infty} \frac{y^k}{k!})$.
Now we need to realize that we can rearrange the terms in the series and multiply terms of the two series since both of them converge absolutely.
Hence $e^x \times e^y = (1 + \sum_{k=1}^{\infty} \frac{x^k}{k!}) \times (1 + \sum_{k=1}^{\infty} \frac{y^k}{k!}) = 1 + (x+y) + (\frac{x^2 + 2xy + y^2}{2!}) + (\frac{x^3 + 3x^2y + 3xy^2 + y^3}{3!}) + \cdots$
Now make use of the binomial theorem to get 
$$e^{x} e^{y} = e^{x+y}$$
PS: Though I have taken care to make sure the line of thought is right, you need to be careful when writing down the argument as to when you can interchange terms in an infinite series, multiply out two infinite series etc etc.