How does a logistic growth curve differ from an exponential growth curve?

If the number of people, y, who have heard a rumor is growing at a rate proportional to how many have heard it, then we get the differential equation y' = ky, where y' = dy/dt. In this case the growth is exponential, and the solution is y = A ekt . The graph keeps getting higher and steeper forever.

Except hold on to his or her beliefs even if there is no evidence for those beliefs.