What’s so “natural” about the base of natural logarithms?

Question 1

There are so many available bases. Why is the strange number $e$ preferred over all else?

Of course one could integrate $\frac{1}x$ and see this. But is there more to the story?

Question 2

Differentiation and integration is precisely why it is considered natural, but not just because
$\displaystyle\int \frac{1}{x} dx=\ln x$

$e^x$ has the two following nice properties

$\frac{d}{dx} e^x=e^x$

$\int e^x dx=e^x+c$

If we looked at $a^x$ instead, we would get:

$\frac {d} {dx} a^x= \frac{d}{dx} e^{x\ln(a)}=\ln(a) \cdot a^x$

$\int a^x dx= \int e^{x\ln(a)} dx=\frac{a^x}{\ln(a)}+c$

So $e$ is vital to the integration and differentiation of exponentials.

Answer