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Exponential growth is the inverse of logarithmic growth. Not all cases of growth at an always increasing rate are instances of exponential growth. For example the function () = grows at an ever increasing rate, but is much slower than growing exponentially.
Relative growth rate (RGR) is growth rate relative to size - that is, a rate of growth per unit time, as a proportion of its size at that moment in time. It is also called the exponential growth rate, or the continuous growth rate.
One equation used to analyze biological exponential growth uses the birth and death rates in a population. If, in a hypothetical population of size N, the birth rates (per capita) are represented as b and death rates (per capita) as d, then the increase or decrease in N during a time period t will be
The growth equation for exponential populations is = where e is Euler's number, a universal constant often applicable in logistic equations, and r is the intrinsic growth rate.
The doubling time is a characteristic unit (a natural unit of scale) for the exponential growth equation, and its converse for exponential decay is the half-life. As an example, Canada's net population growth was 2.7 percent in the year 2022, dividing 72 by 2.7 gives an approximate doubling time of about 27 years.
r = the population growth rate, which Ronald Fisher called the Malthusian parameter of population growth in The Genetical Theory of Natural Selection, [2] and Alfred J. Lotka called the intrinsic rate of increase, [3] [4] t = time. The model can also be written in the form of a differential equation: =
The derivative (rate of change) of the exponential function is the exponential function itself. More generally, a function with a rate of change proportional to the function itself is expressible in terms of the exponential function. This derivative property leads to exponential growth or exponential decay.
The equation for exponential mass growth rate in plant growth analysis is often expressed as: = Where: M(t) is the final mass of the plant at time (t). M 0 is the initial mass of the plant. RGR is the relative growth rate. RGR can then be written as: