By - DioriteLover
It is the unique family of solutions to the remarkably simple differential equation
y' = cy
meaning that the instantaneous rate at which the amount of stuff is increasing (y') is always directly proportional (with constant c) to how much stuff is already there (y).
1. If the number of bacteria doubles every hour, then the instantaneous rate of change of the bacteria is equal to (current number of bacteria) per hour.
2. If an element has a 30% chance of radioactively decaying in any given year, then in a sample of those particles (when averaged over the trillions of trillions of them in your collection) will grow at the instantaneous of rate -0.3 \* (current number of particles) per year.
3. If a bank is offering you a 1% APR compounded continuously, then every cent in your account is "producing" new cents at the instantaneous rate of .01 \* (current number of cents in your account) per year.
In all of these cases, you have a finely-divisible collection of goods (bacteria, particles, cents) for which each division is growing or shrinking at the same constant rate, independently of the existence (or lack thereof) of the others. So in aggregate, the whole collection is growing at that rate as well, relative to its current size.
Compounding growth is compounding growth. It applies to population, ecology, economics, and many more areas...but so does addition, multiplication, etc. It's a standalone mathematical function, not unique to one study.
Basically, the exponential function is the most basic example of how the way a function grows can be related to its current value, in that it's rate of growth is directly proportional to its current value.
What do I mean by that? Well, take interest. How much you're making at any given moment directly depends on how much you have at that moment. The number of people being born/dying at any one time depends on the population size. An object in a gravitational field starts accelerating faster and faster the closer it gets to the center of gravity.
These kinds of relationships are (in a trite and over-simplified sense) what physics is all about, and more generally what continuously varying dynamic systems are all about. It's no surprise the exponential function, which demonstrates the simplest of these types of relationships, comes up often. This is especially true considering that whenever math comes up in a field, we try to make it as simple as possible (because otherwise it's often intractable, and a problem for computers to solve).
In Mathematics, the exponential function is actually even more fundamental than that, in that it's also the 'simplest' type of smooth cyclic motion, but that's a conversation for another time.