*A*=*A*_{0}e^{kt}.

*k*= ln(*A*/*A*_{0})/*t*;*t*= ln(*A*/*A*_{0})/*k*.

You can replace e with any other valid base (2, 10, whatever),
so long as you change *k* to match
(but then *k* is no longer the relative growth rate).
A different choice of the base can make the correct value of *k*
either more or less obvious.
For example, if a quantity doubles in size every *H* years,
then its size after *t* years
is

*A*=*A*_{0}2^{t/H}.

*A*=*A*_{0}2^{−t/h}.

If an object is placed in an environment at constant temperature,
then 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* between the object and its environment
will undergo exponential decay.
If *u* is the temperature of the object
and *T* is the temperature of its environment,
then *u* − *T* is the quantity *A*
in the general formula for exponential growth and decay,
with *u*_{0} − *T*
in place of *A*_{0}:

*u*−*T*= (*u*_{0}−*T*)e^{kt}.

*u*=*T*+ (*u*_{0}−*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.
If *P* is the population and *c* is its carrying capacity,
then *P*/(*c* − *P*)
is the *A* in the general formula for exponential growth and decay,
with *P*_{0}/(*c* − *P*_{0})
in place of *A*_{0}:

*P*/(*c*−*P*) =*P*_{0}/(*c*−*P*_{0}) e^{kt}.

*P*=*c**P*_{0}e^{kt}/(*P*_{0}e^{kt}+*c*−*P*_{0}).

*P*=*c*/(1 +*a*e^{−kt}).

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This web page was written by Toby Bartels, last edited on 2022 December 16. Toby reserves no legal rights to it.

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