The growth curve

You will write three small functions for the logistic model of the lesson.

The idea. The cells after $t$ hours are $\dfrac{K}{1 + (K/N_0 - 1) e^{-rt}}$. The time to reach half of $K$ is $\ln(K/N_0 - 1)/r$. In the log phase, the number doubles at a steady pace.

Your tasks.

1. Finish cells. It gives the number of cells after hours, from the start, the limit and the rate.
2. Finish time_to_half. It gives the hours that are needed to reach half of the limit.
3. Finish doublings. It gives how many times a number doubles when it goes from a to b.