*A*=*P*e^{kt},

For example, if a quantity doubles in size every *H* years,
then its size after *t* years
is

*A*=*P*2^{t/H},

*A*=*P*2^{−t/h}.

If an object is placed in an environment at constant temperature,
it will cool down or heat up to reach the environment's temperature.
This temperature will neither grow nor decay exponentially;
but according to Isaac Newton's **law of cooling and heating**,
the *difference in temperature* will undergo exponential decay:

*A*−*T*= (*P*−*T*)e^{kt},

*A*=*T*+ (*P*−*T*)e^{kt}.

Exponential decay is one thing,
but exponential growth forever is unrealistic.
In the model of **logistic growth**,
there is a **carrying capacity**
beyond which a population cannot grow.
In this case, there is still an exponential growth,
but it is *the ratio of the population to the remaining capacity*
that grows exponentially:

*A*/(*C*−*A*) =*P*/(*C*−*P*) ⋅ e^{kt},

*A*=*C**P*e^{kt}÷ (*C*−*P*+*P*e^{kt}).

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This web page was written between 2012 and 2016 by Toby Bartels, last edited on 2016 May 12. Toby reserves no legal rights to it.

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