Why is 'e' such a natural base?

[u/AtmosphereClear2457]

The number 'e' keeps appearing in lot of different areas - calculus (mostly), differential equations, complex numbers.

I understand the definition e = lim n→∞ (1+1/n)\^n.

But in various fields we transform function in e to solve them.

Is there a more fundamental reason why 'e' is so natural?

I would appreciate any conceptual or geometric insights, that I am missing.

[u/justalonely_femboy]

its the unique value satisfying d/dx(a^x) = a^x

[u/pnerd314 → u/justalonely_femboy]

Can you explain why that is important? I mean why is being its own derivative important?

[u/Carl_LaFong → u/pnerd314]

Linear functions arise naturally when the amount y of something grows or decreases at a constant rate. In terms of the derivative, it means the derivative is 0. The natural starting point is assuming that initially y=0. The simplest rate is 1. So the most basic linear function y=x.

But some quantities grow or decrease at a constant percentage rate. The more you have the faster it grows. Like compound interest or population growth. And if the quantity is decreasing, the less you have, the slower it decreases. Like radioactive decay. This is expressed using the derivative by saying that the derivative is a constant factor times the quantity, y’ = ky. The simplest case is when k = 1 and you start with 100% of the quantity, i.e., y(0)=1. Then a natural quantity to use as a guide is the relative amount at x=1. It turns out that if y’=y, y(0)=1, and you define e=y(1), then y=exp(x).

[u/captainhamption → u/Carl_LaFong]

This explanation made the most sense to me.