Francis Oliver

2022-10-22

Why do we use the form $f(t)=a{e}^{kt}$ for exponential growth and decay?

Why will we include the ${e}^{k}$?

would not or not it's simpler to definitely use $f(t)=a{p}^{t}$ wherein p is the proportion growth according to time.

Is there a reason why the convention is to apply $f(t)=a{e}^{kt}$?

Why will we include the ${e}^{k}$?

would not or not it's simpler to definitely use $f(t)=a{p}^{t}$ wherein p is the proportion growth according to time.

Is there a reason why the convention is to apply $f(t)=a{e}^{kt}$?

Taxinov

Beginner2022-10-23Added 18 answers

The choice of base is arbitrary, but the primary reason is likely that the defining equation for the system looks like

$$\frac{df}{dt}=kf.$$

This means that the rate of growth of the population is proportional (proportionality k) to the population at a given time. The solution is $f(t)=a{e}^{kt}$. It can be re-expressed as you said, but there is no benefit.

$$\frac{df}{dt}=kf.$$

This means that the rate of growth of the population is proportional (proportionality k) to the population at a given time. The solution is $f(t)=a{e}^{kt}$. It can be re-expressed as you said, but there is no benefit.

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