Years for the population to exceed a threshold

Statement

A city of $50\,000$ inhabitants grows by $2{.}5\%$ per year. After how many years will it exceed $75\,000$ inhabitants?

Solution

From $50\,000\,(1{.}025)^{n} = 75\,000$ it follows $1{.}025^{n} = 1{.}5$. $n = \frac{\ln 1{.}5}{\ln 1{.}025} = \frac{0{.}405465}{0{.}024693} \approx \boxed{16{.}42\ \text{years}}.$ Since $n$ must be an integer, the threshold is exceeded during the $17\text{th}$ year.