Exponential growth and decay
Some examples
The key property of exponential functions is that the rate of
growth (or decay) is proportional to how much is already there. As a
result, the following realworld situations (and others!) are modeled
by exponential functions:
 The population of a colony of bacteria can double every 20
minutes, as long as there is enough space and food. The more bacteria
you already have, the more new bacteria you get. This is modeled by
the function $P(t) = P_0 2^{t/20}$, where $P_0$ is the number of
bacteria you start with and $t$ is the time, measured in
minutes.
 The amount of money in a bank account grows exponentially, since
the amount of interest you earn is proportional to the amount of money
you have. If your annual interest rate is $r$, then the law of
compound interest says that $A(t) = A_0 (1+r)^t$, where $t$ is the
elapsed time in years and $A_0$ is the amount of money you start
with.
 In radioactive decay, a certain fraction of the atoms decay every
second, so the rate of shrinkage is proportional to how much is
there. The law is $Y(t) = Y_0 2^{t/\tau}$, where the constant
$\tau$ is called the halflife of the material.
The number $e$
The exact proportionality constant for the function $a^x$ depends
on the number $a$. When $a=2$, the constant is around 0.69. When
$a=3$, the constant is around 1.1. There is a special number,
called $e$, for which the
proportionality constant is exactly 1. That is, the rate at
which $e^x$ grows is exactly $e^x$. The numerical value
of $e$ is approximately 2.718281828.
