Exponential growth

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

$${\displaystyle x_{t}=x_{0}(1+r)^{t}}$$

where x₀ is the value of x at time 0. The growth of a bacterial

Terms like "exponential growth" are sometimes incorrectly interpreted as "rapid growth". Indeed, something that grows exponentially can in fact be growing slowly at first.^[1][2]

Examples

This section needs additional citations for verification. Please help improve this article by adding citations to reliable sources in this section. Unsourced material may be challenged and removed. (August 2013) (Learn how and when to remove this template message)

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.^[3]
• 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

• dielectric breakdown
  of the material.
• 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."^[4]
• 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.^[5] See also rule of 72.
• 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

• Ray Kurzweil
  .)
• In
  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
  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.^[7] 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.^[6][8]

A quantity x depends exponentially on time t if

$${\displaystyle x(t)=a\cdot b^{t/\tau }}$$

$${\displaystyle x(0)=a\,,}$$

$${\displaystyle x(t+\tau )=a\cdot b^{\frac {t+\tau }{\tau }}=a\cdot b^{\frac {t}{\tau }}\cdot b^{\frac {\tau }{\tau }}=x(t)\cdot b\,.}$$

If τ > 0 and b > 1, then x has exponential growth. If τ < 0 and b > 1, or τ > 0 and 0 < b < 1, then x has exponential decay.

Example: If a species of bacteria doubles every ten minutes, starting out with only one bacterium, how many bacteria would be present after one hour? The question implies a = 1, b = 2 and τ = 10 min.

$${\displaystyle x(t)=a\cdot b^{t/\tau }=1\cdot 2^{t/(10{\text{ min}})}}$$

$${\displaystyle x(1{\text{ hr}})=1\cdot 2^{(60{\text{ min}})/(10{\text{ min}})}=1\cdot 2^{6}=64.}$$

After one hour, or six ten-minute intervals, there would be sixty-four bacteria.

Many pairs (b, τ) of a

Thus the law of exponential growth can be written in different but mathematically equivalent forms, by using a different base. The most common forms are the following:

$${\displaystyle x(t)=x_{0}\cdot e^{kt}=x_{0}\cdot e^{t/\tau }=x_{0}\cdot 2^{t/T}=x_{0}\cdot \left(1+{\frac {r}{100}}\right)^{t/p},}$$

Parameters (negative in the case of exponential decay):

• The growth constant k is the
  continuously compounded return, or force of interest
  .
• The e-folding time τ is the time it takes to grow by a factor e.
• The doubling time T is the time it takes to double.
• The percent increase r (a dimensionless number) in a period p.

The quantities k, τ, and T, and for a given p also r, have a one-to-one connection given by the following equation (which can be derived by taking the natural logarithm of the above):

$${\displaystyle k={\frac {1}{\tau }}={\frac {\ln 2}{T}}={\frac {\ln \left(1+{\frac {r}{100}}\right)}{p}}}$$

If p is the unit of time the quotient t/p is simply the number of units of time. Using the notation t for the (dimensionless) number of units of time rather than the time itself, t/p can be replaced by t, but for uniformity this has been avoided here. In this case the division by p in the last formula is not a numerical division either, but converts a dimensionless number to the correct quantity including unit.

A popular approximated method for calculating the doubling time from the growth rate is the

${\displaystyle T\simeq 70/r}$

Graphs comparing doubling times and half lives of exponential growths (bold lines) and decay (faint lines), and their 70/t and 72/t approximations. In the SVG version, hover over a graph to highlight it and its complement.

Reformulation as log-linear growth

If a variable x exhibits exponential growth according to ${\displaystyle x(t)=x_{0}(1+r)^{t}}$, then the log (to any base) of x grows linearly over time, as can be seen by taking logarithms of both sides of the exponential growth equation:

$${\displaystyle \log x(t)=\log x_{0}+t\cdot \log(1+r).}$$

This allows an exponentially growing variable to be modeled with a log-linear model. For example, if one wishes to empirically estimate the growth rate from intertemporal data on x, one can linearly regress log x on t.

Differential equation

The exponential function ${\displaystyle x(t)=x_{0}e^{kt}}$ satisfies the linear differential equation:

$${\displaystyle {\frac {dx}{dt}}=kx}$$

${\displaystyle x(0)=x_{0}}$

The differential equation is solved by direct integration:

$${\displaystyle {\begin{aligned}{\frac {dx}{dt}}&=kx\\[5pt]{\frac {dx}{x}}&=k\,dt\\[5pt]\int _{x_{0}}^{x(t)}{\frac {dx}{x}}&=k\int _{0}^{t}\,dt\\[5pt]\ln {\frac {x(t)}{x_{0}}}&=kt.\end{aligned}}}$$

$${\displaystyle x(t)=x_{0}e^{kt}.}$$

In the above differential equation, if k < 0, then the quantity experiences exponential decay.

For a nonlinear^{[disambiguation needed]} variation of this growth model see logistic function.

Other growth rates

In the long run, exponential growth of any kind will overtake linear growth of any kind (that is the basis of the

growth, that is, for all α:

$${\displaystyle \lim _{t\to \infty }{\frac {t^{\alpha }}{ae^{t}}}=0.}$$

There is a whole hierarchy of conceivable growth rates that are slower than exponential and faster than linear (in the long run). See Degree of a polynomial § Computed from the function values.

Growth rates may also be faster than exponential. In the most extreme case, when growth increases without bound in finite time, it is called hyperbolic growth. In between exponential and hyperbolic growth lie more classes of growth behavior, like the hyperoperations beginning at tetration, and ${\displaystyle A(n,n)}$, the diagonal of the Ackermann function.

Logistic growth

In reality, initial exponential growth is often not sustained forever. After some period, it will be slowed by external or environmental factors. For example, population growth may reach an upper limit due to resource limitations.

Studies show that human beings have difficulty understanding exponential growth. Exponential growth bias is the tendency to underestimate compound growth processes. This bias can have financial implications as well.[11]

Below are some stories that emphasize this bias.

Rice on a chessboard

According to an old legend, vizier Sissa Ben Dahir presented an Indian King Sharim with a beautiful handmade chessboard. The king asked what he would like in return for his gift and the courtier surprised the king by asking for one grain of rice on the first square, two grains on the second, four grains on the third, etc. The king readily agreed and asked for the rice to be brought. All went well at first, but the requirement for 2ⁿ⁻¹ grains on the nth square demanded over a million grains on the 21st square, more than a million million (a.k.a. trillion) on the 41st and there simply was not enough rice in the whole world for the final squares. (From Swirski, 2006)^[12]

The

Water lily

French children are offered a riddle, which appears to be an aspect of exponential growth: "the apparent suddenness with which an exponentially growing quantity approaches a fixed limit". The riddle imagines a water lily plant growing in a pond. The plant doubles in size every day and, if left alone, it would smother the pond in 30 days killing all the other living things in the water. Day after day, the plant's growth is small, so it is decided that it won't be a concern until it covers half of the pond. Which day will that be? The 29th day, leaving only one day to save the pond.^[13][12]

See also

References