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?
On
You can apply the same principle as in my previous solution. What you need is the growth rate in a month. Call the rate per month $e^s$, then
$$(e^s)^{12}=e^r=1.115$$
because after 12 months you of course have 1 year. You can solve this to give you
$$e^s=\sqrt[12]{1.115}=1.009112$$
You can work it out from here.
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 $y$.