指数增长

变量x指数地依赖时间t，若

${\displaystyle x(t)=a\cdot b^{\frac {t}{\tau }}\,}$ 

其中常数a是x的初始值,

${\displaystyle x(0)=a\,,}$ 

并且，常数b是正的增长率，τ为x增长b倍所需时间：

${\displaystyle x(t+\tau )=x(t)\cdot b\,.}$ 

若τ > 0且b > 1，则x为指数增长。若τ < 0且b > 1，或τ > 0且0 < b < 1，则x为指数衰减。

指数函数${\displaystyle \scriptstyle x(t)=ae^{kt}}$ 满足线性微分方程：

${\displaystyle \!\,{\frac {dx}{dt}}=kx}$ 

则称t时刻x的增长率与函数值x(t)成正比，且初值为：

${\displaystyle x(0)=a.\,}$ 

对于${\displaystyle \scriptstyle a>0}$ 微分方程可以使用分离变量法求解：

${\displaystyle {\frac {dx}{dt}}=kx}$ 

${\displaystyle \Rightarrow {\frac {dx}{x}}=k\,dt}$ 

${\displaystyle \Rightarrow \int {\frac {dx}{x}}=\int k\,dt}$ 

${\displaystyle \Rightarrow \ln x=kt+{\text{constant}}\,.}$ 

考虑到给定初值：

${\displaystyle \ln x=kt+\ln a\,}$ 

${\displaystyle \Rightarrow x=ae^{kt}\,}$ 

这种解法对于${\displaystyle \scriptstyle a\leq 0}$ 同样适用。

对于该增长模型的非线性变体，请参考Logistic函数。

• 摩尔定律
• 国际象棋盘与麦粒问题

• Meadows, Donella H., Dennis L. Meadows, Jørgen Randers, and William W. Behrens III. (1972) The Limits to Growth. New York: University Books. ISBN 0-87663-165-0
• Porritt, J. Capitalism as if the world matters, Earthscan 2005. ISBN 1-84407-192-8
• Thomson, David G. Blueprint to a Billion: 7 Essentials to Achieve Exponential Growth, Wiley Dec 2005, ISBN 0-471-74747-5
• Tsirel, S. V. 2004. On the Possible Reasons for the Hyperexponential Growth of the Earth Population. Mathematical Modeling of Social and Economic Dynamics / Ed. by M. G. Dmitriev and A. P. Petrov, pp. 367–9. Moscow: Russian State Social University, 2004.