*Although growth may initially be exponential, the modelled phenomena will eventually enter a region in which previously ignored negative feedback factors become significant (leading to a logistic growth model) or other underlying assumptions of the exponential growth model, such as continuity or instantaneous feedback, break down.*

*Although growth may initially be exponential, the modelled phenomena will eventually enter a region in which previously ignored negative feedback factors become significant (leading to a logistic growth model) or other underlying assumptions of the exponential growth model, such as continuity or instantaneous feedback, break down.*

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 .

In this way, each increase in the exponent by a full interval can be seen to increase the previous total by another five percent.

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.

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).

For example, if one wishes to empirically estimate the growth rate from intertemporal data on x, one can linearly regress log x on t.

The exponential function showing that x experiences exponential growth.

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: where k = 0 corresponds to r = 0 and to τ and T being infinite.

## Comments How To Solve Exponential Growth Problems

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