We observe that the elementary logistic differential equation dP/dt=(1-P/M)kP
may be solved by first changing the variable to R=(M-P)/P. This reduces the
logistic differential equation to the simple linear differential equation
dR/dt=-kR, which can be solved without using the customary but slightly more
elaborate methods applied to the original logistic DE. The resulting solution
in terms of R can be converted by simple algebra to the familiar sigmoid
expression involving P. A biological argument is given for introducing logistic
growth via the simpler DE for R. It is also shown that the sigmoid P may be
written in terms of the hyperbolic tangent by a simple translation that is also
motivated by a biological argument.Comment: 5 pages AMSLaTeX, 2 figure