Exponential

Exponential growth (including exponential decay) occurs when the growth rate of a mathematical function is proportional to the function's current value. In the case of a discrete domain of definition with equal intervals it is also called geometric growth or geometric decay (the function values form a geometric progression).

With exponential growth of a positive value its rate of increase steadily increases, or in the case of exponential decay, its rate of decrease steadily decreases.

Exponential growth is said to follow an exponential law; the simple-exponential growth model is known as the Malthusian growth model. For any exponentially growing quantity, the larger the quantity gets, the faster it grows. An alternative saying is 'The rate of growth is directly proportional to the present size'. The relationship between the size of the dependent variable and its rate of growth is governed by a strict law of the simplest kind: direct proportion. It is proved in calculus that this law requires that the quantity is given by the exponential function, if we use the correct time scale. This explains the name.

Examples

• Biology.
  • The number of microorganisms in a culture broth will grow 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.
  • A virus ([for example SARS, West Nile 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).
  • Many responses of living beings to stimuli, including human perception, are logarithmic responses, which are the inverse of exponential responses; the loudness and frequency of sound are perceived logarithmically, even with very faint stimulus, within the limits of perception. This is the reason that exponentially increasing the brightness of visual stimuli is perceived by humans as a linear increase, rather than an exponential increase. This has survival value. Generally it is important for the organisms to respond to stimuli in a wide range of levels, from very low levels, to very high levels, while the accuracy of the estimation of differences at high levels of stimulus is much less important for survival.
• 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 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.
• Multi-level marketing

Exponential increases are promised to appear in each new level of a starting member's downline as each subsequent member recruits more people.

• 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.).
  • In computational complexity theory, computer algorithms of exponential complexity require an exponentially increasing amount of resources (e.g. time, computer memory) for only a constant increase in problem size. So for an algorithm of time complexity 2^x, 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. E.g. 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.
  • Internet traffic growth.
• Investment. Compound interest at a constant interest rate provides exponential growth of the capital. See also rule of 72

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
• 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

• Exponent calculator — One of the best ways to see how exponents work is to simply try different examples. 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 8.5 min
• Dr. Albert Bartlett: Arithmetic, Population and Energy — sreaming video and audio 58 min