# Determining how long a body has been dead using the number e

I have recently seen a quote about determining how long a body has been dead:

“Dead bodies lose heat exponentially, and therefore e can be used in an appropriate equation to determine how long individuals have been dead” (Calvin Clawson, Mathematical Mysteries, 1996).

Does anyone have an idea about such equation?

• I guess $e^{-t}$.
– Alex
Mar 2, 2015 at 13:43

I don't know the specifics, but the equation will be something of the form $$H=Ae^{-kt}$$ where $H$ is heat, $t$ is time since death and $A,k$ are (known) constants. Then, if you measure the heat $H$, you can find the time $t$ since death by: \begin{align} H&=Ae^{-kt}\\ \frac{H}{A}&=e^{-kt}\\ -kt&=\log\left(\frac{H}{A}\right)\\ t&=\frac{-1}{k}\log\left(\frac{H}{A}\right)\\ \end{align} I disgree that this method really involves the number $e$, though. Indeed, we could replace $e$ with any exponent $a$ and get the same thing, since: $$a^{-kx}=\left(e^{\log(a)}\right)^{-kx}=e^{-k\log(a)x}=e^{-\hat{k}x}$$ where $\hat{k}=k\log(a)$. In other words, passing from $e$ to another exponent just changes the constant $k$.
$e$ is certainly the most natural exponent, though, for reasons that will become clear to you in the course of your mathematical education.