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Exponential Growth


Exponential growth is the increase in a quantity N according to the law

 N(t)=N_0e^(lambdat)
(1)

for a parameter t and constant lambda (the analog of the decay constant), where e^x is the exponential function and N_0=N(0) is the initial value. 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

 (dN)/(dt)=lambdaN,
(2)

which can be rearranged to

 (dN)/N=lambdadt.
(3)

Integrating both sides then gives

 ln(N/(N_0))=lambdat,
(4)

and exponentiating both sides yields the functional form (1).

A much more antiquated term for population growth modeled according to an exponential equation is the so-called Malthusian equation, a result of a 1798 philosophical text by Thomas Malthus which investigated population dynamics under the assumption that the growth of the human population obeys a sort of exponential growth.


See also

Exponential Function, Exponential Decay, Growth, Law of Growth, Malthusian Equation, Malthusian Parameter, Population Growth Explore this topic in the MathWorld classroom

Portions of this entry contributed by Christopher Stover

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References

Malthus, T. R. "An Essay on the Principle of Population." 1798. http://www.econlib.org/library/Malthus/malPop.html.

Referenced on Wolfram|Alpha

Exponential Growth

Cite this as:

Stover, Christopher and Weisstein, Eric W. "Exponential Growth." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ExponentialGrowth.html

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