For interest Compounded continuously
The formula for calculating continuously compounded interest is expressed as
A = Pe^rt
where
A is the final amount after t years
P is the principal or initial amount
r is the interest rate
t is the number of years
From the information given,
final amount = 2 x initial amount
Thus,
A = 2P
r = 13/100 = 0.13
By substituting these values into the formula, we have
2P = Pe^0.13t
Dividing both sides by P, we have
2P/P = P/Pe^0.13t
2 = e^0.13t
Taking the natural log of both sides, we have
ln 2 = ln e^0.13t
Recall the following rules of logarithm
lna^b = blna
lne = 1
Thus, we have
ln 2 = 0.13tln e
ln 2 = 0.13t
0.13t = ln 2
Dividing both sides by 0.13, we have
0.13t/0.13 = ln 2/0.13
t = 5.33
It will take 5.33 years for the investment to double in value.