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Exponential growth
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Exponential growth

In mathematics, a quantity that grows exponentially is one that grows at a rate proportional to its size. Anything that grows by the same percentage every year (or every month, day, hour etc.) is growing exponentially. For example, if the average number of offspring of each individual (or couple) in a population remains constant, the growth is proportional to the number of individuals. Such an exponentially growing population grows three times as fast in individuals per year when there are six million individuals, as it does when there are two million. Bank accounts with fixed-rate compound interest grow exponentially provided there are no deposits, withdrawals or service charges.

Table of contents
1 Misnomer
2 Technicalities
3 Examples of exponential growth

Misnomer

The phrase exponential growth is also incorrectly used by persons not versed in quantitative matters to mean merely surprisingly fast growth (a potential malapropism). In fact, a population can grow exponentially but at a very slow rate (as the fission process in a nuclear power plant), and can grow surprisingly fast without growing exponentially. Indeed, the logistic function grows approximately exponentially when it is growing very slowly, but nowhere near exponentially when it is growing fastest.

Technicalities

If we call x this quantity, the rate of change dx/dt obeys by definition the differential equation:

where α > 0 is the constant of proportionality (the average number of offspring in the case of the population). (See logistic function for a simple correction of this model growth where α is not constant). The solution to this equation is the exponential function x(t)=C exp(αt), whence the name of the associated growth. C here is an arbitrary constant, determined by the initial size of the population.

In the long run, exponential growth of any kind will however overtake linear growth of any kind (the basis of the Malthusian catastrophe) as well as any polynomial growth, i.e., for all α:

There is a whole hierarchy of conceivable growth laws that are sub-exponential and also super-linear, and of course growth faster than exponential is also possible. The linear and exponential models are merely simple candidates but are those of greatest occurrence in nature.

Examples of exponential growth

See also: exponential decay, bacterial growth, logistic curve, arthrobacter, exponential algorithm, exponential function, asymptotic notation, Rule of 72