e and its applications In math when people want to model population growth or radioactive decay we use exponential functions. In many cases, we use base $e$. My question is, what is the purpose of using base $e$ rather than some other base? 
 A: The exponential decay phenomena in nature do not involve $e$, but other basis of arbitrary value (more precisely, there are arbitrary time constants). In a way, $e$ does not exist in nature, neither in the discharge of condensers, nor in the decay of radioactive materials, nor on the spirals of sunflowers, nor in the growth of compound loan interests.
$e$ is the preferred base of mathematicians because it is convenient, in the sense that it is the only base for which $$\left(b^x\right)'=b^x.$$
For other bases, a conversion factor appears. This is the very same reason why radians are used to measure angles.
On the opposite, the base $10$ that we use for our numeration system is deeply grounded in human anatomy, and base $2$ is pretty useful in the theory of information and computer science.
A: For starters, sometimes people do use other bases. In radioactivity, $2$ is often used, giving meaning to the term half-life.
To answer your question, there is no reason to pick any particular base unless it is useful. Since 2 and $e$ are useful, we use them. If another number is useful for a specific problem having to do with exponential growth or decay, people use it. The reason that $e$ is useful is because $e^{\lambda x}$ is the solution to $y' = \lambda y$, a fundamental differential equation for this type of problem.
A: The number $e$ appears naturally as the limit of the compounding interest formula. That is $$\left( 1 + \frac{r}{n} \right)^n \simeq e^r$$ when $n$ is large. This makes $e$ a good choice for modeling systems with exponential growth.

In order to expand on the above. The compounding interest formula is one of the routes which led to the discovery of $e$. (The other route was through the natural logarithm, which is related to the area underneath the hyperbola $y=1/x$.) Liebniz was among the earlier mathematicians to study $e$, and he used $b$ to represent this number. I believe the proof demonstrating $$\lim_{n\to\infty}\left( 1 + \frac{r}{n} \right)^n = \sum_{n=0}^\infty \frac{r^n}{n!}$$ may be due to him (I would have to consult with some references I don't have handy at the moment).
There is an excellent book on the history of $e$, named appropriately "e: The story of a number". It's a book written for laymen, and I thoroughly enjoyed reading it.
