When to use $f(x)=Ce^{kx}$ vs $f(x)=Ca^{kx}$?

Question

(Where 'e' is 2,71828...) When do you really want to use one over the other? What properties do they have?

Answer 1

This depends on what you want to focus attention to. If the case of interest on some ammount of money, the interesting part is how much you get in a year, so letting $x$ stand for the number of years, it would be fitting to use the function $$ C\cdot p^x $$ In this case $p$ is the total interest accumulated over a single year.

If you're working with radioactive materials, the interesting part is how long time it takes to expend half of its matter (the material's half-life). A function of radiation (or total matter left, the two are proportional) would then be $$ C\cdot 2^{-\frac{1}{t}\cdot x} $$ Here $t$ is the half-life of the material, in the same unit of time as $x$.

For a purely mathematical use, the function $$ C\cdot e^{kx} $$ behaves nicest out of all options, so if you're not focusing on a specific property of what the equation is describing, then it is usually the best. All in all, it is only a matter of convenience anyway, so if it's not worth it to rewrite it to the latter form, then you shouldn't do it in my opinion.

Answer 2

You can absorb the base into $k$. $Ce^{kx \ln a}=Ca^kx$ so you can express the same things. Sometimes one base is more convenient than another. $2,e,$ and $10$ are popular. $e$ has the advantage that $\frac d{dx}e^x=e^x$ where the others pick up a factor of the log of the base.