A common question is why e (2.71828...) is so special. Why not 2, 3.7 or some other number as the base of growth?

First off, e was discovered, not chosen. Think of the speed of light, c. It wasn't originally decided to be 299,792,458 m/s -- we did experiments and realized under ideal, universal conditions (a vacuum), this was the fastest light could move1.

Let's look at growth and ask under ideal, universal conditions, what's the fastest something can possibly grow? Ideal, universal assumptions would be:

• Growing by the unit rate (100%)
• Growing for the unit time (1 time period)
• Growing perfectly, without any delay (continuous)

Turning these assumptions into a formula, we get:

If we actually use the formula (using large values of n for more accuracy) we get e = 2.718281828459...

Objection: But 13.74x can model exponential growth just like ex can!

Sure. But what assumptions did you make to get 13.74? They probably weren't "unit rate, unit time, perfectly compounded". (You can pick k as the speed of light through Kool Aid too -- but why?)

Arguably, 2n is also universal ("the discrete e"), because you have zero compounding (n is an integer like 0, 1, 2, 3). Instead of perfectly continuous, it's perfectly non-continuous (discrete), and we take growth step-by-step.

So, I'd say either 2n (in discrete systems) or ex (in continuous systems) are "universal".

Objection: But things can grow faster than ex, which is just 2.71828x -- what about 13.74x?

What is it with you and 13.74? Yes, you can beat ex in an exponential footrace, if you use a rate more than 100%. ​13.74x is really ​[eln(13.74)]x. Because ln(13.74) ~ 2.6, you are assuming a 260% continuous interest rate, more than the 100% ​ex uses. (Alternatively, you can grow for 260% of the unit time period that ​ex uses.)

Related:

1. Funny enough, in 1983 c was decreed to be 299,792,458 m/s by redefining the length of a meter. Similarly, you could decide that e is a clean "10" in your base-e number system.