Exponential growth is the increase in a quantity 
according to the law
 |
(1)
|
for a parameter 
and a positive constant
growth rate 
,
where 
is the exponential function and 
is the initial value. With the same positive
initial value, 
gives exponential decay, while 
gives a constant quantity.
Exponential growth is common in physical processes such as population
growth in the absence of predators or resource restrictions (where a slightly
more general form is known as the law of growth).
Exponential growth also occurs as the limit of discrete
processes such as compound interest.
Exponential growth is described by the first-order ordinary differential equation
 |
(2)
|
which can be rearranged to
 |
(3)
|
Integrating both sides then gives
 |
(4)
|
and exponentiating both sides yields the functional form (3).
A much more antiquated term for population growth modeled according to an exponential equation is the so-called Malthusian equation, a result of the philosophical text by Malthus (1798) which investigated population dynamics under the assumption that the growth of the human population obeys a sort of exponential growth.
