The Power of the Exponential

Borne from the notion of self-similar growth, the exponential has far more utility than being just another function.

The exponential is a key concept that helps us understand compounding behaviors like the spread of a disease (credit: Elena Mozhvilo on Unsplash)

From pandemics to economics, the exponential growth is a common phrase used to emphasize a dramatic level of change. So how do we understand this notion rigorously?

To begin, we consider a function whose growth is proportional to itself. In calculus, this can be defined as a function whose slope is itself. In other words, we are attempting to solve the following equation:

How is all this related to compound interests? It turns out that hidden in this definition is a continuous recursive behavior. Furthermore, the recursion has applications beyond simple growing behavior: we can compound more abstract things, like rotations, translations, and probabilities.

Exponential as Recursions

To reveal the continuous recursion, we need to dig deeper to the definition of the slope (or alternatively, the growth) of a function. This is defined as taking the limit of a progressive series of approximations:

If we ignore the limit symbol, we can better understand what it means when a function’s slope is itself:

Where we have used the approximate ≈ symbol to indicate approximate behaviors for very small ε. While our discussion…