Why is $e$ so ubiquitous? $e$ is one of the most important numbers in our universe, it is everywhere. When I try to find out why the most common explanation is some reverse-engineered physics or finance problem. But these are just one off examples of why $e$ _is_ important, they fall short of illuminating the origin of its significance. What I'm looking for is some fundamental definition of $e$ that explains it's significance and omnipresence, something akin to $\pi$ relating circumference and diameter. Thanks, I hope I'm being clear. EDIT*: A common saying is "$e$ is the most natural base." What does that mean?

\begin{align} \frac d {dx} 10^x & = ( 10^x\cdot\text{constant}) \approx 10^x \cdot (2.3) \\\\[10pt] \frac d {dx} 2^x & = ( 2^x \cdot \text{constant}) \approx 2^x\cdot (0.693) \end{align} etc. It is easy to show that the derivative of an exponential function is a constant multiple of the same exponential function.

Only when the bas is $e$ is the "constant" equal to $1$.

The fact that $x\mapsto e^x$ is its own derivative accounts for its incessant appearance in the study of differential equations. It also accounts for the fact that the "constant" is the base-$e$ logarithm of the base of the epxonential function.

That's the beginning of the story; there's a lot more to it.

The fact that the "constant" is equal to $1$ only when the base is $e$ is analogous to the fact that in the identity $$ \frac d {dx} \sin x = (\text{constant}\cdot \cos x) $$ the "constant" is $1$ only when **radians** are used rather than some other unit.