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

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

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Do you mean $Ce^{kx}$ versus $Ca^{kx}$? –  Thomas Andrews Dec 11 '12 at 19:34
There is an argument for sometimes using $a=10$, we have some intuition about powers of $10$. In Computer Science, there is often a good argument for using $a=2$. Sometimes it is convenient to use $a=1/10$ or $a=1/2$ to avoid negative exponents. –  André Nicolas Dec 11 '12 at 19:35

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.
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.