# How do you determine the multiplier for exponential growth and decay?

For an exponential model $P = f \left(t\right) = {P}_{0} {e}^{k t}$, where ${P}_{0}$ is the initial value (at time $t = 0$). You can find the value of $k$ if you know another data point by using logarithms.
For example, if $P = f \left(t\right) = 100 {e}^{k t}$ and you know $f \left(3\right) = 200$, then $200 = 100 {e}^{3 k}$ so that ${e}^{3 k} = 2$, $3 k = \ln \left(2\right)$, and k=\frac{1}{3}ln(2)\approx 0.231=23.1%. This represents an "instantaneous relative rate of change".
In general, if you are given the data point $\left({t}_{1} , {P}_{1}\right)$, where ${t}_{1}$ is not zero (in addition to the data point $\left(0 , {P}_{0}\right)$), then ${P}_{1} = {P}_{0} {e}^{k {t}_{1}}$ so that $k = \setminus \frac{1}{{t}_{1}} \ln \left(\setminus \frac{{P}_{1}}{{P}_{0}}\right)$.