Exponential growth
exponential growth, exponential growth formulaExponential growth is a phenomenon that occurs when the growth rate of the value of a mathematical function is proportional to the function's current value, resulting in its growth with time being an exponential function Exponential decay occurs in the same way when the growth rate is negative In the case of a discrete domain of definition with equal intervals, it is also called geometric growth or geometric decay, the function values forming a geometric progression In either exponential growth or exponential decay, the ratio of the rate of change of the quantity to its current size remains constant over time
The formula for exponential growth of a variable x at the growth rate r, as time t goes on in discrete intervals that is, at integer times 0, 1, 2, 3, , is
x t = x 0 1 + r t =x_1+r^where x0 is the value of x at time 0 This formula is transparent when the exponents are converted to multiplication For instance, with a starting value of 50 and a growth rate of r = 5% = 005 per interval, the passage of one interval would give 50 105 1 , or simply 50×105; two intervals would give 50 105 2 , or simply 50×105×105; and three intervals would give 50 105 3 , or simply 50×105×105×105 In this way, each increase in the exponent by a full interval can be seen to increase the previous total by another five percent The order of multiplication does not change the result based on the associative property of multiplication
Since the time variable, which is the input to this function, occurs as the exponent, this is an exponential function
Contents
 1 Examples
 2 Basic formula
 3 Reformulation as loglinear growth
 4 Differential equation
 5 Difference equation
 6 Other growth rates
 7 Limitations of models
 8 Exponential stories
 81 Rice on a chessboard
 82 Water lily
 9 See also
 10 References and footnotes
 101 Sources
 11 External links
Examples
Bacteria exhibit exponential growth under optimal conditions
This section needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed August 2013 Learn how and when to remove this template message  The number of microorganisms in a culture will increase exponentially until an essential nutrient is exhausted Typically the first organism splits into two daughter organisms, who then each split to form four, who split to form eight, and so on Because exponential growth indicates constant growth rate, it is frequently assumed that exponentially growing cells are at a steadystate However, cells can grow exponentially at a constant rate while remodelling their metabolism and gene expression
 A virus for example SARS, or smallpox typically will spread exponentially at first, if no artificial immunization is available Each infected person can infect multiple new people
 Human population, if the number of births and deaths per person per year were to remain at current levels but also see logistic growth For example, according to the United States Census Bureau, over the last 100 years 1910 to 2010, the population of the United States of America is exponentially increasing at an average rate of one and a half percent a year 15% This means that the doubling time of the American population depending on the yearly growth in population is approximately 50 years
 Physics
 Avalanche breakdown within a dielectric material A free electron becomes sufficiently accelerated by an externally applied electrical field that it frees up additional electrons as it collides with atoms or molecules of the dielectric media These secondary electrons also are accelerated, creating larger numbers of free electrons The resulting exponential growth of electrons and ions may rapidly lead to complete dielectric breakdown of the material
 Nuclear chain reaction the concept behind nuclear reactors and nuclear weapons Each uranium nucleus that undergoes fission produces multiple neutrons, each of which can be absorbed by adjacent uranium atoms, causing them to fission in turn If the probability of neutron absorption exceeds the probability of neutron escape a function of the shape and mass of the uranium, k > 0 and so the production rate of neutrons and induced uranium fissions increases exponentially, in an uncontrolled reaction "Due to the exponential rate of increase, at any point in the chain reaction 99% of the energy will have been released in the last 46 generations It is a reasonable approximation to think of the first 53 generations as a latency period leading up to the actual explosion, which only takes 3–4 generations"
 Positive feedback within the linear range of electrical or electroacoustic amplification can result in the exponential growth of the amplified signal, although resonance effects may favor some component frequencies of the signal over others
 Economics
 Economic growth is expressed in percentage terms, implying exponential growth For example, US GDP per capita has grown at an exponential rate of approximately two percent since World War 2
 Finance
 Compound interest at a constant interest rate provides exponential growth of the capital See also rule of 72
 Pyramid schemes or Ponzi schemes also show this type of growth resulting in high profits for a few initial investors and losses among great numbers of investors
 Computer technology
 Processing power of computers See also Moore's law and technological singularity Under exponential growth, there are no singularities The singularity here is a metaphor, meant to convey an unimaginable future The link of this hypothetical concept with exponential growth is most vocally made by transhumanist Ray Kurzweil
 In computational complexity theory, computer algorithms of exponential complexity require an exponentially increasing amount of resources eg time, computer memory for only a constant increase in problem size So for an algorithm of time complexity 2x, if a problem of size x = 10 requires 10 seconds to complete, and a problem of size x = 11 requires 20 seconds, then a problem of size x = 12 will require 40 seconds This kind of algorithm typically becomes unusable at very small problem sizes, often between 30 and 100 items most computer algorithms need to be able to solve much larger problems, up to tens of thousands or even millions of items in reasonable times, something that would be physically impossible with an exponential algorithm Also, the effects of Moore's Law do not help the situation much because doubling processor speed merely allows you to increase the problem size by a constant Eg if a slow processor can solve problems of size x in time t, then a processor twice as fast could only solve problems of size x+constant in the same time t So exponentially complex algorithms are most often impractical, and the search for more efficient algorithms is one of the central goals of computer science today
Basic formula
A quantity x depends exponentially on time t if
x t = a ⋅ b t / τ \,where the constant a is the initial value of x,
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 =a\cdot b^\cdot b^=xt\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 =1\cdot 2^/10 x 1 hr = 1 ⋅ 2 6 = 64 =1\cdot 2^=64After one hour, or six tenminute intervals, there would be sixtyfour bacteria
Many pairs b, τ of a dimensionless nonnegative 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 eg e or 2, the growth rate is given by the nonzero time τ For any nonzero 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 , \cdot e^=x_\cdot e^=x_\cdot 2^=x_\cdot \left1+\right^,where x0 expresses the initial quantity x0
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 efolding 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 onetoone 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 ==\right\,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, ie T ≃ 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 complementReformulation as loglinear growth
If a variable x exhibits exponential growth according to x t = x 0 1 + r t 1+r^ , 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 +t\cdot \log1+rThis allows an exponentially growing variable to be modeled with a loglinear 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 satisfies the linear differential equation:
d x d t = k x =kxsaying that the growth rate of x at time t is proportional to the value of xt, and it has the initial value
x 0The differential equation is solved by direct integration:
d x d t = k x =kx ⇒ d x x = k d t =k\,dt ⇒ ∫ x 0 x t d x x = k ∫ 0 t d t ^=k\int _^\,dt ⇒ ln x t x 0 = k t =ktso that
⇒ x t = x 0 e k t \,For a nonlinear variation of this growth model see logistic function
Difference equation
The difference equation
x t = a ⋅ x t − 1 =a\cdot x_has solution
x t = x 0 ⋅ a t , =x_\cdot a^,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, ie, for all α:
lim t → ∞ t α a e t = 0 \over ae^=0There 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 Inbetween exponential and and hyperbolic growth lie more classes of growth behavior, like the hyperoperations beginning at tetration, and A n , n , the diagonal of the Ackermann function
In the above differential equation, if k < 0, then the quantity experiences exponential decay
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 rateExponential stories
Rice on a chessboard
See also: Wheat and chessboard problemAccording to an old legend, vizier Sissa Ben Dahir presented an Indian King Sharim with a beautiful, handmade chessboard The king asked what he would like in return for his gift and the courtier surprised the king by asking for one grain of rice on the first square, two grains on the second, four grains on the third etc The king readily agreed and asked for the rice to be brought All went well at first, but the requirement for 2 n − 1 grains on the nth square demanded over a million grains on the 21st square, more than a million million aka trillion on the 41st and there simply was not enough rice in the whole world for the final squares From Swirski, 2006
The second half of the chessboard is the time when an exponentially growing influence is having a significant economic impact on an organization's overall business strategy
Water lily
French children are told a story in which they imagine having a pond with water lily leaves floating on the surface The lily population doubles in size every day and if left unchecked will smother the pond in 30 days, killing all the other living things in the water Day after day the plant seems small and so it is decided to leave it to grow until it halfcovers the pond, before cutting it back They are then asked on what day halfcoverage will occur This is revealed to be the 29th day, and then there will be just one day to save the pond From Meadows et al 1972
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
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:101016/jcelrep201403057 ISSN 22111247
 ^ 2010 Census Data “US Census Bureau” 20 Dec 2012 Internet Archive: http://webarchiveorg/web/20121220035511/http://2010censusgov/2010census/data/indexphp
 ^ Sublette, Carey "Introduction to Nuclear Weapon Physics and Design" Nuclear Weapons Archive Retrieved 20090526
 ^ a b Porritt, Jonathan 2005 Capitalism: as if the world matters London: Earthscan p 49 ISBN 1844071928
Sources
 Meadows, Donella H, Dennis L Meadows, Jørgen Randers, and William W Behrens III 1972 The Limits to Growth New York: University Books ISBN 0876631650
 Porritt, J Capitalism as if the world matters, Earthscan 2005 ISBN 1844071928
 Swirski, Peter Of Literature and Knowledge: Explorations in Narrative Thought Experiments, Evolution, and Game Theory New York: Routledge ISBN 0415420601
 Thomson, David G Blueprint to a Billion: 7 Essentials to Achieve Exponential Growth, Wiley Dec 2005, ISBN 0471747475
 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
 Exponent calculator — This calculator enables you to enter an exponent and a base number and see the result
 Exponential Growth Calculator — This calculator enables you to perform a variety of calculations relating to exponential consumption growth
 Understanding Exponential Growth — video clip 85 min
 Growth in a Finite World – Sustainability and the Exponential Function — Presentation
 Dr Albert Bartlett: Arithmetic, Population and Energy — streaming video and audio 58 min
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