# If Emery has $1400 to invest at 11% per year compounded monthly, how long will it be before he has$2300. If the compounding is continuous, how long will it be?

Apr 10, 2018

Emery will need $55$ months.

#### Explanation:

Using the compound interest formula: $A = P {\left(1 + r\right)}^{n}$

$2300 = 1400 {\left(1 + \frac{0.11}{12}\right)}^{n}$

$\frac{23}{14} = {\left(1 + \frac{0.11}{12}\right)}^{n}$ divide both sides by $1400$

$\log \left(\frac{23}{14}\right) = \log {\left(1 + \frac{0.11}{12}\right)}^{n}$ take $\log$ of both sides

$\log \left(\frac{23}{14}\right) = n \log \left(1 + \frac{0.11}{12}\right)$ $\log$ law: $\log {x}^{n} = n \log x$

$\log \frac{\frac{23}{14}}{\log} \left(1 + \frac{0.11}{12}\right) = n$ divide both sides by $\log \left(1 + \frac{0.11}{12}\right)$

$n \approx 54.4$