Constant growth rate? Say the population of a city is increasing at a constant rate of 11.5% per year. If the population is currently 2000, estimate how long it will take for the population to reach 3000.
Using the formula given, so far I've figured out how many years it will take (see working below) but how can I narrow it down to the nearest month? 



Let $a=1.115^{1/12}=\sqrt[12]{1.115}$, the twelfth root of $1.115$. Then 
$$1.115^x=(a^{12})^x=a^{12x}\;,$$
and $12x$ is the number of months that have gone by. Thus, if you can solve $a^y=1.5$, $y$ will be the desired number of months. Without logarithms the best that you’ll be able to do is find the smallest integer $y$ such that $a^y\ge 1.5$.
By my calculation $a\approx1.009112468437$. You could start with $a^{36}$ and work up until you find the desired $y$.
