Chapter 6.8, Problem 56SE

Recall that the general form of a logistic equation for a population is given by $P\left(t\right)=\frac{c}{1+a{e}^{-bt}}$ , such that the initial population at time

$t=0$ is $P\left(0\right)=P_0$. Show algebraically that $\frac{c-P\left(t\right)}{P\left(t\right)}=\frac{c-P_0}{P_0}{e}^{-bt}$ .