In the preceding part, we determined the reasonableness of a logistic fit (up to 1940) and estimated the parameters r and K using only the differential equation, not the symbolic solution found in Part 5. Now we see what we can do by using the solution, which we recall has the form
where P0 is the population at whatever time we declare to be time 0. For example, if t = 0 in 1790, then P0 = 3.929. For purposes of this exercise, we will make that choice of starting point and measure all times from 1790.
In Part 5, as a step on the way to the symbolic solution, we saw that the solution would have to satisfy
that is, ln (P / (K - P)) should be a linear function of t with slope r. Thus, given a value of K, we can plot ln (P / (K - P)) against t and see if we get a straight line. (Note that there is no need to approximate rates of change for this type of test.) Since you already have an estimate of K from Part 6, it will not be a surprise if the plot looks pretty straight on the first try.
and t in step 2, make a new estimate of r.