Calculate the time needed for a certain bacterial growth

It has been discovered that a population of bacteria grows according to the following law: If there are nt bacteria at time t then one hour later at time t + 1 there will be

where both - alpha and K - are positive numbers that controls the growth rate and defines capacity, respectively.

**Hints:** Remember to import all the packages. You will need to import numpy and math.

**Example:** Starting with n0 = 100 bacteria and setting α = 0.3 and K = 1010, let us find out when the population exceeds N = 500. Running the simulation, we get the following sequence of population sizes.