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I am looking for interesting applications in everyday life, technology or science of exponential functions of the type:

$$ f\colon \mathbb{R} \to \mathbb{R}, \quad x \mapsto ab^x $$

for the case $a > 0$ and $b > 1$.

I am also interested in other examples which are in some way fascinating or funny for this case.

I don't want any discrete examples or such involving the Euler number $\mathrm{e}$.

Any suggestions for that?

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    $\begingroup$ maybe try the half life decay in radioactivity. $\endgroup$ – mookid Mar 15 '14 at 13:49
  • $\begingroup$ @mookid Yes but thats for the case $b < 1$, it's a decay. I know there are many interesting (quasi-)continous examples with $b < 1$, but I didn't know any really interesting examples for $b > 1$ which are not discrete. $\endgroup$ – Julia Mar 15 '14 at 13:54
  • $\begingroup$ oh I did not see the condition on $b$. $\endgroup$ – mookid Mar 15 '14 at 14:15
  • $\begingroup$ We cannot throw out functions of $e$.$\begin{align}f(x) &= ab^x \\ &= e^(ln(ab^x))\\ &= e^(ln(a) + ln(b^x))\\ &= e^(ln(a) +xln(b)\\ &=e^{c +bx}\end{align}$ $\endgroup$ – Brad S. Mar 15 '14 at 14:49
  • $\begingroup$ Well that's clear. The point is just that I cannot assume that students are familiar with $\mathrm{e}$. $\endgroup$ – Julia Mar 15 '14 at 15:15
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$a>0$ and $b>1$ correspond to exponential growths. Thius kind of formula is used in studies about population, virus, bacteria, ...But you have one case which is very common : make a deposit of $a$ dollars on your bank account and let us suppose that the bank gives you an interest of $r$% per year. After $n$ years, your initial $a$ dollars will become $a \left(1+\frac{r}{100}\right)^n$ dollars.

Now, if your bank is generous and if you are patient, you will be rich. Suppose that you invest today $100$ dollars, that your generous bank gives you a $5$ % interest (give me its name) and that you are ready to wait for $20$ years, then, in year 2034, you will be able to cash $265$ dollars.

Unfortunately, with bacteria and virus, this goes much faster. For example if we start with only one bacteria which can double every hour, by the end of one day we will have almost 17 million bacteria.

Another example is the number of cell phones. They increased just as bacteria (slightly slower but very fast).

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    $\begingroup$ Dollars, bacteria and cell phones are all discrete. $\endgroup$ – Brad S. Mar 15 '14 at 14:52
  • $\begingroup$ @BradS. Almost everything can be made discrete or continuous, I bet; it just depends how far is the observer from the object. But you are correct ! I just wanted to take simple, practical examples and avoid physics (by the way : particles are discrete, just as atoms or molecules, don't you agree ?). Cheers. $\endgroup$ – Claude Leibovici Mar 15 '14 at 14:56
  • $\begingroup$ Yes. of course. It is actually a pretty interesting question. I'm sitting here with my morning coffee trying to think of an example...I'm thinking maybe the mass of a tree or some such similar growth. $\endgroup$ – Brad S. Mar 15 '14 at 15:01
  • $\begingroup$ Enjoy your coffee, think about it and come back, please. I think that you initiated a very interesting discussion. Have a nice sunday. Cheers. $\endgroup$ – Claude Leibovici Mar 15 '14 at 15:03
  • $\begingroup$ Thanks. Your compound interest example is excellent. I remember banks (and savings & Loans) in the 1980's advertising "continuous compounding" and "compounded daily". $\endgroup$ – Brad S. Mar 15 '14 at 15:24

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