Editing Exponential growth (section)

{{Short description|Growth of quantities at rate proportional to the current amount}}
{{Use dmy dates|date=February 2024}}
[[File:Exponential.svg|thumb|right|300px|The graph illustrates how exponential growth (green) eventually surpasses both linear (red) and cubic (blue) growth.

{{legend|red|Linear growth}} {{legend|blue|[[Polynomial|Cubic growth]]}} {{legend|green|Exponential growth}}]]

'''Exponential growth''' occurs when a quantity grows as an [[exponential function]] of time. The quantity grows at a rate [[direct proportion|directly proportional]] to its present size. For example, when it is 3 times as big as it is now, it will be growing 3 times as fast as it is now.

In more technical language, its instantaneous [[Rate (mathematics)#Of change|rate of change]] (that is, the [[derivative]]) of a quantity with respect to an independent variable is [[proportionality (mathematics)|proportional]] to the quantity itself. Often the independent variable is time. Described as a [[Function (mathematics)|function]], a quantity undergoing exponential growth is an [[Exponentiation#Power functions|exponential function]] of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as [[quadratic growth]]). Exponential growth is [[Inverse function|the inverse]] of [[logarithmic growth]].

Not all cases of growth at an always increasing rate are instances of exponential growth. For example the function <math display=inline> f(x) = x^3 </math> grows at an ever increasing rate, but is much slower than growing exponentially. For example, when <math display=inline> x=1,</math> it grows at 3 times its size, but when <math display=inline> x=10 </math> it grows at 30% of its size. If an exponentially growing function grows at a rate that is 3 times is present size, then it always grows at a rate that is 3 times its present size. When it is 10 times as big as it is now, it will grow 10 times as fast.

If the constant of proportionality is negative, then the quantity decreases over time, and is said to be undergoing [[exponential decay]] instead. In the case of a discrete [[Domain of a function|domain]] of definition with equal intervals, it is also called '''geometric growth''' or '''geometric decay''' since the function values form a [[geometric progression]].

The formula for exponential growth of a variable {{mvar|x}} at the growth rate {{mvar|r}}, as time {{mvar|t}} goes on in discrete intervals (that is, at integer times&nbsp;0,&nbsp;1,&nbsp;2,&nbsp;3,&nbsp;...), is

<math display="block">x_t = x_0(1+r)^t</math>

where {{math|''x''<sub>0</sub>}} is the value of {{mvar|x}} at time 0. The growth of a bacterial [[Colony (biology)|colony]] is often used to illustrate it. One bacterium splits itself into two, each of which splits itself resulting in four, then eight, 16, 32, and so on. The amount of increase keeps increasing because it is proportional to the ever-increasing number of bacteria. Growth like this is observed in real-life activity or phenomena, such as the spread of virus infection, the growth of debt due to [[compound interest]], and the spread of [[viral video]]s. In real cases, initial exponential growth often does not last forever, instead slowing down eventually due to upper limits caused by external factors and turning into [[logistic curve|logistic growth]].

Terms like "exponential growth" are sometimes incorrectly interpreted as "rapid growth". Indeed, something that grows exponentially can in fact be growing slowly at first.<ref>{{Cite news|url=https://www.nytimes.com/2019/03/04/opinion/exponential-language-math.html|title=Opinion &#124; Stop Saying 'Exponential.' Sincerely, a Math Nerd.| first=Manil| last=Suri|newspaper=The New York Times |date=4 March 2019}}</ref><ref>{{Cite web|url=https://science.howstuffworks.com/dictionary/astronomy-terms/10-scientific-words-using-wrong.htm|title=10 Scientific Words You're Probably Using Wrong|date=11 July 2014| website=HowStuffWorks}}</ref>