Instead of half-life would exp-life be easier to calculate? Specifically if we know how long it takes $t_0$ for $\frac{1}{e}$ of the principle amount and we know the principle amount we would be able to form the exponential equation $P_{0}e^{rt}$ where $t_{0}r=-\ln(e)=-1$. Just a thought I had while teaching was that it is more natural to talk about an Exp-life, instead of a half-life, does anyone agree? 
 A: I like the following property - "exp-life" could as well stand for "expected life". Expected time before a given atom decays is indeed its exp-life. $$\int_0^\infty e^{-\frac t T}dt = T$$
But when you are interested in changes to overall quantity, half-life still seems easier to visualise.
A: I teach half-life in calculus class, and it is pretty easy if you use this form of the equation:
$$y = P_0\left(\frac 12\right)^{t/h}$$
where $y$ and $P_0$ are the final and initial amounts, both with the same unit, $t$ is the elapsed time from time zero, and $h$ is the half-life, with the same unit as $t$.
The equation is easy to understand, ties directly in with half-life, and can be solved for $t$ by using logarithms to base one-half. Mainly, the students understand this clearly, before they learn about derivatives. After a while, I use this to show how natural logarithms can be used for any exponential equation.
I find this works well, better than some other approaches I tried. Its main weakness is in taking the derivative, which results in a more messy expression than if you use $e$ as the base. It still works well overall and seems more natural than an exp-life would be. I tried that concept once, without that particular name, and it did not work as well.
