Exponential growth

In mathematics, a quantity that grows exponentially is one that grows at a rate proportional to its size. This means 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.

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 (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 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.
• Electroengineering
• 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).
• 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. See also rule of 72
• 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 $T=Ae^\{-kt\},$ where T is temperature, t is time, and, A and k > 0 are constants, is an example of exponential decay.