**Gist**

Exponential growth is a process that increases quantity over time at an ever-increasing rate. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself.

**Details**

Exponential growth is a process that increases quantity over time at an ever-increasing rate. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth). Exponential growth is the inverse of logarithmic growth.

If the constant of proportionality is negative, then the quantity decreases over time, and is said to be undergoing exponential decay instead. In the case of a discrete domain of definition with equal intervals, it is also called geometric growth or geometric decay since the function values form a geometric progression.

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

where x0 is the value of x at time 0. The growth of a bacterial colony is often used to illustrate it. One bacterium splits itself into two, each of which splits itself resulting in four, then eight, 16, 32, and so on. The amount of increase keeps increasing because it is proportional to the ever-increasing number of bacteria. Growth like this is observed in real-life activity or phenomena, such as the spread of virus infection, the growth of debt due to compound interest, and the spread of viral videos. In real cases, initial exponential growth often does not last forever, instead slowing down eventually due to upper limits caused by external factors and turning into logistic growth.

Terms like "exponential growth" are sometimes incorrectly interpreted as "rapid growth". Indeed, something that grows exponentially can in fact be growing slowly at first.

**Examples**

Bacteria exhibit exponential growth under optimal conditions.

**Biology**

The number of microorganisms in a culture will increase exponentially until an essential nutrient is exhausted, so there is no more of that nutrient for more organisms to grow. 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 steady-state. However, cells can grow exponentially at a constant rate while remodeling their metabolism and gene expression.

A virus (for example COVID-19, or smallpox) typically will spread exponentially at first, if no artificial immunization is available. Each infected person can infect multiple new people.

**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), 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 4.6 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.

**Finance**

Compound interest at a constant interest rate provides exponential growth of the capital.

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 science**

Processing power of computers.

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 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 increases the feasible 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 today.

**Internet phenomena**

Internet contents, such as internet memes or videos, can spread in an exponential manner, often said to "go viral" as an analogy to the spread of viruses. With media such as social networks, one person can forward the same content to many people simultaneously, who then spread it to even more people, and so on, causing rapid spread. For example, the video Gangnam Style was uploaded to YouTube on 15 July 2012, reaching hundreds of thousands of viewers on the first day, millions on the twentieth day, and was cumulatively viewed by hundreds of millions in less than two months.

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