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

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

Wouldn't it be easier to simply use $f(t)=ap^{t}$ where $p$ is the percentage increase per time.

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

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Shouldn't those be $f(t)$ rather than $x$? – Karolis Juodelė Jan 20 '13 at 19:24
Oops, you're right. I'll change it right now. – Linksku Jan 20 '13 at 19:26
It is not widely known, but growth and decay problems led to the name of an important modern area of study, Grothendieck K-theory. See en.wikipedia.org/wiki/Grothendieck_group and en.wikipedia.org/wiki/K-theory – Will Jagy Jan 20 '13 at 21:18
P.S. say them out loud... – Will Jagy Jan 20 '13 at 21:27

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 $f(t)=ae^{kt}$. It can be re-expressed as you said, but there is no benefit.