My question is: Suppose that I have $\$2,500$ in an investment account. I want this to grow to $\$5,000$.
Approximately how long it would take to do this if my account earns $3.5\%$ compounded annually $$ FV=PV(1+i)^n \\ 5000=2500(1+.035)^n \\ 5000=2500(1.035)^n $$ I need help with this question. It's one of my business mathematics questions. I'm confused as to how to divide out the $n$ or how I would arrive at $n$.
The general technique when the $n$ is in the exponent is to use $\log$ and then use the rule $\log(x)^n=n \log(x)$. \begin{align*} 5000 &=2500(1.035)^n \\ 5000/2500 &=(1.035)^n \\ \log(5000/2500) &= \log((1.035)^n) \\ \log(5000/2500) &=n \log(1.035) \\ n &=\frac{\log(5000/2500)}{\log(1.035)} \approx 20.15 \end{align*}
A quick and dirty way to get the number of years to double your money is to use the rule of $72$ http://en.wikipedia.org/wiki/Rule_of_72 which says that it will take about $72/3.5 \approx 20.57$ years.
So, if you need to calculate the number of years , $y$, it takes for an initial value, $P$, to accumulate its interest to $F$, where the interest is $i$ in % per annum, your formula is:
$$y = \frac{\log(F/P)}{\log(1 + (i / 100))}$$