When using exponential growth why is the form $a \cdot e^{ct}$ used instead of $a\cdot b^t$?

Question

My maths book is not very forthcoming. My guess would be that it is because when you use the form $a \cdot e^{ct}$ is easier to differentiate to see the rate of growth.

Answer 1

You can convert between the two. $\log(a^t) = t\log a = \log(e^{t\log a})$, so $a^t = e^{t\log a}$. Using base $e$ just happens to be nice because you don't have to carry around the extra $\log a$ everywhere when you differentiate or integrate. $e$ is in some sense the ideal base for the purposes of differentiation.

The take-home message is: exponential growth is exponential growth is exponential growth.