How do logistic and exponential growth differ?

Logistic is limited growth, and exponential is unbounded growth.

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 #e^(kt)#. The graph keeps getting higher and steeper forever.

Examples of exponential growth are compound interest and bacteria growth, assuming an infinite amount of money or fuel.

But there are finitely many people B in the world, currently B = #7.2 * 10^9#, so eventually people hearing the rumor will have already heard it, and the growth slows down. This is logistic growth. The curve starts out like an exponential, but the growth rate starts decreasing (at y = B/2), and eventually approaches its horizontal asymptote of y = B at time goes on. And on. The inflection point is at the fastest rate of increase.

Logistic growth is seen in "learning curves", internet popularity, and (did you hear) rumors! - - - \dansmath strikes again/