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

1 Answer
Apr 23, 2015

For an exponential model P=f(t)=P_{0}e^{kt}P=f(t)=P0ekt, where P_{0}P0 is the initial value (at time t=0t=0). You can find the value of kk if you know another data point by using logarithms.

For example, if P=f(t)=100e^{kt}P=f(t)=100ekt and you know f(3)=200f(3)=200, then 200=100e^{3k}200=100e3k so that e^{3k}=2e3k=2, 3k=ln(2)3k=ln(2), and k=\frac{1}{3}ln(2)\approx 0.231=23.1%k=13ln(2)0.231=23.1%. This represents an "instantaneous relative rate of change".

In general, if you are given the data point (t_{1},P_{1})(t1,P1), where t_{1}t1 is not zero (in addition to the data point (0,P_{0})(0,P0)), then P_{1}=P_{0}e^{kt_{1}}P1=P0ekt1 so that k=\frac{1}{t_{1}}ln(\frac{P_{1}}{P_{0}})k=1t1ln(P1P0).