Good article in general, but when I learned this it made sense at least to me.

The first logarithm you encounter is the "common" log, to the base 10, because it's "the number of digits in your number".

But 10 has no intrinsic meaning. If we had two more fingers, we'd be using base 12. 😀

Now, a function grows exponentially exactly if its rate of change is proportional to the value of the function, so you say that the population of the world increases at a rate of 1.2% a year, say, or inflation is 5%.

This is just the simplest actually interesting differential equation, f'(x) = k f(x), and the simplest case is of course f'(x) = f(x) which has the solution f(x) = e ** x, and as you point out, this leads to the integration of 1/x being the logarithm to the base e, which makes this the mathematically natural choice for an exponential base.

This is a bit less convincing now I wrote it out, but the literature seems to have other people saying the same.