In mathematics, a quantity that grows exponentially is one that grows at a rate proportional to its value. For example, if the average number of offspring of each individual (or couple) in a population remains constant, the growth is proportional to the number of individuals. Such an exponentially growing population grows three times as fast in individuals per year when there are six million individuals, as it does when there are two million. Also, a snowball rolling downhill grows exponentially with time since when it is twice as big it gathers snow twice as fast.

If we call x this quantity, the rate of change dx/dt obeys by definition the differential equation:

where is the constant of proportionality (the average number of offspring in the case of the population). (See the logistic map for a simple correction of this model growth where α is not constant). The solution to this equation is the exponential function x(t)=C·exp(αt), whence the name of the associated growth. C here is an arbitrary constant, determined by the initial size of the population.

The phrase exponential growth is also a misnomer used by persons unaware of quantitative matters to mean merely surprisingly fast growth. In fact, a population can grow exponentially but at a very slow rate (as the fission process in a nuclear power plant), and can grow surprisingly fast without growing exponentially.

In the long run, exponential growth of any kind will however overtake linear growth of any kind (the basis of the Malthusian catastrophe) as well as any polynomial growth, i.e., for all α:

There is a whole hierarchy of conceivable growth laws that are sub-exponential and also super-linear, and of course growth faster than exponential is also possible. The linear and exponential models are merely simple candidates but are those of greatest occurrence in nature.

Examples of Exponential Growth

• Investing. The effect of compound interest over many years has a substantial effect on savings and a person's ability to retire.
• Biology.
  • Bacteria in a culture dish will grow exponentially until the available food is exhausted.
  • A new virus (SARS, West Nile, smallpox) will spread exponentially. Each infected person can infect multiple new people.
  • Human population.
• An atomic bomb. Each uranium atom that undergoes fission produces neutrons, which in turn split more uranium atoms. If the mass of uranium is sufficent, the number of neutrons increases exponentially.
• A nuclear power plant. Same as above but this time the fission process (also called divergence of the reactor) is controlled so that the growth, while exponential, is very slow.
• Processing power of computers. See also Moore's Law.

See also: exponential decay, bacterial growth, logistic curve, arthrobacter, exponential algorithm, exponential function, asymptotic notation Rule of 72