What's so "natural" about the base of natural logarithms? 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?

If you consider all exponential equations $a^x$, they all have $y$-intercept $(0,1)$. If you wanted to specify an archetypal exponential equation to refer to as you work through Calculus, a natural choice would be to choose the one whose tangent line at $(0,1)$ has slope 1. The equation $e^x$ is the unique exponential equation with that property.
