Contents 1 Examples 2 Basic formula 3 Reformulation as log-linear growth 4 Differential equation 5 Difference equation 6 Other growth rates 7 Limitations of models 8 Exponential stories 8.1 Rice on a chessboard 8.2 Water lily 9 See also 10 References and footnotes 10.1 Sources 11 External links

Basic formula A quantity x depends exponentially on time t if x ( t ) = a ⋅ b t / τ {\displaystyle x(t)=a\cdot b^{t/\tau }} where the constant a is the initial value of x, x ( 0 ) = a , {\displaystyle x(0)=a\,,} the constant b is a positive growth factor, and τ is the time constant—the time required for x to increase by one factor of b: x ( t + τ ) = a ⋅ b t + τ τ = a ⋅ b t τ ⋅ b τ τ = x ( t ) ⋅ b . {\displaystyle x(t+\tau )=a\cdot b^{\frac {t+\tau }{\tau }}=a\cdot b^{\frac {t}{\tau }}\cdot b^{\frac {\tau }{\tau }}=x(t)\cdot b\,.} If τ > 0 and b > 1, then x has exponential growth. If τ < 0 and b > 1, or τ > 0 and 0 < b < 1, then x has exponential decay. Example: If a species of bacteria doubles every ten minutes, starting out with only one bacterium, how many bacteria would be present after one hour? The question implies a = 1, b = 2 and τ = 10 min. x ( t ) = a ⋅ b t / τ = 1 ⋅ 2 ( 60  min ) / ( 10  min ) {\displaystyle x(t)=a\cdot b^{t/\tau }=1\cdot 2^{(60{\text{ min}})/(10{\text{ min}})}} x ( 1  hr ) = 1 ⋅ 2 6 = 64. {\displaystyle x(1{\text{ hr}})=1\cdot 2^{6}=64.} After one hour, or six ten-minute intervals, there would be sixty-four bacteria. Many pairs (b, τ) of a dimensionless non-negative number b and an amount of time τ (a physical quantity which can be expressed as the product of a number of units and a unit of time) represent the same growth rate, with τ proportional to log b. For any fixed b not equal to 1 (e.g. e or 2), the growth rate is given by the non-zero time τ. For any non-zero time τ the growth rate is given by the dimensionless positive number b. Thus the law of exponential growth can be written in different but mathematically equivalent forms, by using a different base. The most common forms are the following: x ( t ) = x 0 ⋅ e k t = x 0 ⋅ e t / τ = x 0 ⋅ 2 t / T = x 0 ⋅ ( 1 + r 100 ) t / p , {\displaystyle x(t)=x_{0}\cdot e^{kt}=x_{0}\cdot e^{t/\tau }=x_{0}\cdot 2^{t/T}=x_{0}\cdot \left(1+{\frac {r}{100}}\right)^{t/p},} where x0 expresses the initial quantity x(0). Parameters (negative in the case of exponential decay): The growth constant k is the frequency (number of times per unit time) of growing by a factor e; in finance it is also called the logarithmic return, continuously compounded return, or force of interest. The e-folding time τ is the time it takes to grow by a factor e. The doubling time T is the time it takes to double. The percent increase r (a dimensionless number) in a period p. The quantities k, τ, and T, and for a given p also r, have a one-to-one connection given by the following equation (which can be derived by taking the natural logarithm of the above): k = 1 τ = ln ⁡ 2 T = ln ⁡ ( 1 + r 100 ) p {\displaystyle k={\frac {1}{\tau }}={\frac {\ln 2}{T}}={\frac {\ln \left(1+{\frac {r}{100}}\right)}{p}}} where k = 0 corresponds to r = 0 and to τ and T being infinite. If p is the unit of time the quotient t/p is simply the number of units of time. Using the notation t for the (dimensionless) number of units of time rather than the time itself, t/p can be replaced by t, but for uniformity this has been avoided here. In this case the division by p in the last formula is not a numerical division either, but converts a dimensionless number to the correct quantity including unit. A popular approximated method for calculating the doubling time from the growth rate is the rule of 70, i.e. T ≃ 70 / r {\displaystyle T\simeq 70/r} . Graphs comparing doubling times and half lives of exponential growths (bold lines) and decay (faint lines), and their 70/t and 72/t approximations. In the SVG version, hover over a graph to highlight it and its complement.

Reformulation as log-linear growth If a variable x exhibits exponential growth according to x ( t ) = x 0 ( 1 + r ) t {\displaystyle x(t)=x_{0}(1+r)^{t}} , then the log (to any base) of x grows linearly over time, as can be seen by taking logarithms of both sides of the exponential growth equation: log ⁡ x ( t ) = log ⁡ x 0 + t ⋅ log ⁡ ( 1 + r ) . {\displaystyle \log x(t)=\log x_{0}+t\cdot \log(1+r).} This allows an exponentially growing variable to be modeled with a log-linear model. For example, if one wishes to empirically estimate the growth rate from intertemporal data on x, one can linearly regress log x on t.

Differential equation The exponential function x ( t ) = x ( 0 ) e k t {\displaystyle x(t)=x(0)e^{kt}} satisfies the linear differential equation: d x d t = k x {\displaystyle \!\,{\frac {dx}{dt}}=kx} saying that the change per instant of time of x at time t is proportional to the value of x(t), and x(t) has the initial value x ( 0 ) . {\displaystyle x(0).} The differential equation is solved by direct integration: d x d t = k x d x x = k d t ∫ x ( 0 ) x ( t ) d x x = k ∫ 0 t d t ln ⁡ x ( t ) x ( 0 ) = k t . {\displaystyle {\begin{aligned}{\frac {dx}{dt}}&=kx\\[5pt]{\frac {dx}{x}}&=k\,dt\\[5pt]\int _{x(0)}^{x(t)}{\frac {dx}{x}}&=k\int _{0}^{t}\,dt\\[5pt]\ln {\frac {x(t)}{x(0)}}&=kt.\end{aligned}}} so that x ( t ) = x ( 0 ) e k t {\displaystyle x(t)=x(0)e^{kt}} In the above differential equation, if k < 0, then the quantity experiences exponential decay. For a nonlinear variation of this growth model see logistic function.

Difference equation The difference equation x t = a ⋅ x t − 1 {\displaystyle x_{t}=a\cdot x_{t-1}} has solution x t = x 0 ⋅ a t , {\displaystyle x_{t}=x_{0}\cdot a^{t},} showing that x experiences exponential growth.

Other growth rates In the long run, exponential growth of any kind will overtake linear growth of any kind (the basis of the Malthusian catastrophe) as well as any polynomial growth, i.e., for all α: lim t → ∞ t α a e t = 0. {\displaystyle \lim _{t\rightarrow \infty }{t^{\alpha } \over ae^{t}}=0.} There is a whole hierarchy of conceivable growth rates that are slower than exponential and faster than linear (in the long run). See Degree of a polynomial#The degree computed from the function values. Growth rates may also be faster than exponential. In the most extreme case, when growth increases without bound in finite time, it is called hyperbolic growth. In between exponential and hyperbolic growth lie more classes of growth behavior, like the hyperoperations beginning at tetration, and A ( n , n ) {\displaystyle A(n,n)} , the diagonal of the Ackermann function.

Limitations of models Exponential growth models of physical phenomena only apply within limited regions, as unbounded growth is not physically realistic. Although growth may initially be exponential, the modelled phenomena will eventually enter a region in which previously ignored negative feedback factors become significant (leading to a logistic growth model) or other underlying assumptions of the exponential growth model, such as continuity or instantaneous feedback, break down. Further information: Limits to Growth, Malthusian catastrophe, and Apparent infection rate

See also Accelerating change Albert Allen Bartlett Arthrobacter Asymptotic notation Bacterial growth Bounded growth Cell growth Exponential algorithm EXPSPACE EXPTIME Hausdorff dimension Hyperbolic growth Information explosion Law of accelerating returns List of exponential topics Logarithmic growth Logistic curve Malthusian growth model Menger sponge Moore's law Multiplicative calculus Stein's law

References and footnotes ^ Slavov, Nikolai; Budnik, Bogdan A.; Schwab, David; Airoldi, Edoardo M.; van Oudenaarden, Alexander (2014). "Constant Growth Rate Can Be Supported by Decreasing Energy Flux and Increasing Aerobic Glycolysis". Cell Reports. 7 (3): 705–714. doi:10.1016/j.celrep.2014.03.057. ISSN 2211-1247. PMC 4049626 . PMID 24767987.  ^ 2010 Census Data, "U.S. Census Bureau", 20 Dec 2012, Internet Archive: https://web.archive.org/web/20121220035511/http://2010.census.gov/2010census/data/index.php ^ Sublette, Carey. "Introduction to Nuclear Weapon Physics and Design". Nuclear Weapons Archive. Retrieved 2009-05-26.  ^ a b Porritt, Jonathan (2005). Capitalism: as if the world matters. London: Earthscan. p. 49. ISBN 1-84407-192-8.  Sources 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 Swirski, Peter. Of Literature and Knowledge: Explorations in Narrative Thought Experiments, Evolution, and Game Theory. New York: Routledge. ISBN 0-415-42060-1 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.

External links Growth in a Finite World – Sustainability and the Exponential Function — Presentation Dr. Albert Bartlett: Arithmetic, Population and Energy — streaming video and audio 58 min Retrieved from "https://en.wikipedia.org/w/index.php?title=Exponential_growth&oldid=819325244" Categories: Ordinary differential equationsExponentialsMathematical modelingHidden categories: Articles needing additional references from August 2013All articles needing additional referencesAll articles with unsourced statementsArticles with unsourced statements from August 2013Pages using div col with deprecated parameters