Question 1140196
the continuous compounding formula is f = p * e^(r * n)


f is the future value
p is the present value
r is the interest rate per time period.
n is the number of time periods.


when r = k, the formula becomes f = p * e ^ (k * n)


when f = 2358.79 and p = 2000 and n = 3 years, the formula becomes:


2358.79 = 2000 * e ^ (3 * k)


divide both sides of this equation by 2000 to get:


2358.79 / 2000 = e ^ (3 * k)


take the natural log of both sides of this equation to get:


ln(2358.79 / 2000) = ln(e ^ (3 * k))


since ln(e ^ (3 * k)) is equal to 3 * k * ln(e) and since ln(e) is equal to 1, the equation becomes:


ln(2358.79 / 2000) = 3 * k


divide both sides of this equation by 3 to get:


ln(2358.79 / 2000) / 3 = k


solve for k to get k = .0550005317


your exponential growth function is:


2358.79 = 2000 * e ^ (.0550005317 * 3).


evaluate this function to get 2358.79 = 2358.79, confirming the solution is correct.


to find the doubling time, the formula becomes:


2 = e ^ (.0550005317 * n)


take the natural log of both sides of this equation to get:


ln(2) = ln(e ^ (.0550005317 * n)


this becomes ln(2) = .0550005317 * n


divide both sides by .0550005317 to get:


ln(2) / .0550005317 = n


solve for n to get n = 12.602555418


replace n in the doubling formula to get:


2 = e ^ (.0550005317 * 12.602555418)


evaluate this function to get 2 = 2, confirming the solution is correct.


note that 2 = e ^ (.0550005317 * n) is the same as 2 = 1 * e ^ (.0550005317 * n)


you future value is 2 and your present value is 1, hence the doubling.