# How to find an equation of exponential decay?

Oct 31, 2014

An equation of exponential decay typically takes the form $A \left(t\right) = A \left(0\right) {e}^{k t}$ where $k < 0$, though sometimes these will be written as $A \left(0\right) {e}^{- k t}$ and have $k > 0$. Either form is acceptable, though some argue that the first form is more accurate, so that is the form that shall be used here. Note that there are several possible sets of information you may be given.

As your question deals specifically with finding the equation (as opposed to the value the function takes at some time $t = m$) you will typically be asked to find either $k$, or more rarely $A \left(0\right)$ (also written sometimes as ${A}_{0}$). In a case where one is being asked to find $k$, typically one has been given the value for ${A}_{0}$ (the initial value of the function at time $t = 0$) and $A \left(m\right)$ (the value of $A \left(t\right)$ at time $t = m$. From this, we can find the value of $k$ and complete our function as follows:

$A \left(m\right) = {A}_{0} {e}^{k m}$
$\frac{A \left(m\right)}{A} _ 0 = {e}^{k m}$
$\ln \left(\frac{A \left(m\right)}{A} _ 0\right) = k m$ (remember that $\ln \left({e}^{x}\right) = x$)
$\ln \left(A \left(m\right)\right) - \ln \left({A}_{0}\right) = k m$
$\frac{\ln \left(A \left(m\right)\right) - \ln \left({A}_{0}\right)}{m} = k$

Then this value of $k$ can be substituted in, and since we know ${A}_{0}$, we can complete our function of $A \left(t\right) = {A}_{0} {e}^{k t}$ (note that $t$ is still our independent variable, $A \left(t\right)$ our dependent variable, and $e$ a constant approx. equal to 2.718282, defined such that $\ln \left(e\right) = 1$

Alternately, if one is tasked with finding $A \left(0\right)$, one has likely already been given $k$ and the value $A \left(m\right)$ of the function at some time $t = m$. In this case, our process is far easier.

$A \left(m\right) = {A}_{0} {e}^{k m}$

Since we have values for $A \left(m\right) , k ,$ and $m$ (which is our value for $t$ at this point, recall), and since an exponential function cannot equal 0 unless the base equals 0, we can simply perform division to obtain:

$\frac{A \left(m\right)}{{e}^{k m}} = {A}_{0}$