Models & Qualitative Behavior · Challenge problem

A rumour spreads through a town according to the logistic model $P' = 0.5P\left(1 - \dfrac{P}{K}\right)$ ($t$ in days), where $P$ is the number of people who have heard it. How many days does it take to go from $10\%$ of the town to $90\%$ of the town? Give an exact answer.