# If an account that earns interest compounded continuously takes 12 years to do in value, how long will it take to triple in value?

Dec 8, 2016

We first need the growth factor $g$, the number that we use to multiply the amount with every year.

#### Explanation:

Example: if the yearly interest is 5% then the amount is multiplied by (100%+5%)/(100%)=1,05 every year, so after three years it is multiplied three times, or by
$1.05 \times 1.05 \times 1.05 = {1.05}^{3} = 1.157625$

We need to find $g$ from ${g}^{12} = 2$
$g = \sqrt{2} = {2}^{\frac{1}{12}} = 1.05946 \ldots$

Next we need to solve ${\left(1.05946 \ldots\right)}^{t} = 3$ for $t$ (in years).

Method 1 - Using logs:
$\log {\left(1.05946 \ldots\right)}^{t} = \log 3 \to$
$t \cdot \log 1.05946 \ldots = \log 3 \to$
$t = \frac{\log 3}{\log 1.05946 \ldots} = 19.02 \ldots \to$

Answer : $t = 19.02$ years.

Method 2 - Using the GC:
Set ${Y}_{1} = {1.05946}^{X}$
Set ${Y}_{2} = 3$
$X : 0 \to 24$ (twice the time for doubling)
$Y : 0 \to 6$ (so the horizontal line will be in the middle)