A quantity that grows exponentially (or geometrically) is one whose growth rate is always proportional to its current size. Such growth is said to follow an exponential law. This implies that for any exponentially growing quantity, the larger the quantity gets, the faster it grows. But it also implies that 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.

Biology.

- Microorganisms in a culture dish will grow exponentially, at first, after the first microorganism appears (but then logistically until the available food is exhausted, when growth stops).

- A virus (SARS, West Nile, smallpox) of sufficient infectivity (k > 0) 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 constant.

- 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 smooth (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.

Electrical engineering

- Charging and discharging of capacitors and changes in current in inductors are also exponential growth and decay phenomena. Engineers use a rule of five time constants to estimate when a steady state has been reached.

Computer technology

- Processing power of computers. See also Moore's law and technological singularity (under exponential growth, there are no such singularities).

- 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, then a problem of size x=11 will require 20 seconds, and 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 one. 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+1 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. The effect of compound interest over many years has a substantial effect on savings and a person's ability to retire.

Physics

- Atmospheric pressure decreases exponentially with increasing height above sea level, at a rate of about 12% per 1000m.

- 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.

- Newton's law of cooling where T is temperature, t is time, and, A, D, and k > 0 are constants, is an example of exponential decay.

Multi-level marketing

- Exponential increases appear in each level of a starting member's downline as each subsequent member recruits more people.

**Examples of exponential growth**Biology.

- Microorganisms in a culture dish will grow exponentially, at first, after the first microorganism appears (but then logistically until the available food is exhausted, when growth stops).

- A virus (SARS, West Nile, smallpox) of sufficient infectivity (k > 0) 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 constant.

- 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 smooth (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.

Electrical engineering

- Charging and discharging of capacitors and changes in current in inductors are also exponential growth and decay phenomena. Engineers use a rule of five time constants to estimate when a steady state has been reached.

Computer technology

- Processing power of computers. See also Moore's law and technological singularity (under exponential growth, there are no such singularities).

- 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, then a problem of size x=11 will require 20 seconds, and 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 one. 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+1 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. The effect of compound interest over many years has a substantial effect on savings and a person's ability to retire.

Physics

- Atmospheric pressure decreases exponentially with increasing height above sea level, at a rate of about 12% per 1000m.

- 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.

- Newton's law of cooling where T is temperature, t is time, and, A, D, and k > 0 are constants, is an example of exponential decay.

Multi-level marketing

- Exponential increases appear in each level of a starting member's downline as each subsequent member recruits more people.