How long will it take to double your money if you earn 5%, compounded continuously?
find t such that
e^(.05t) = 2
note
ln [ e^x ] = x
To determine how long it will take to double your money when earning 5% interest, compounded continuously, we can use the formula for continuous compounding:
A = P * e^(rt)
Where:
A = the final amount (the original amount doubled)
P = the principal amount (the initial amount of money)
e = Euler's number (approximately equal to 2.71828)
r = the annual interest rate (in decimal form)
t = the time (in years)
Since we want to double the initial amount, A = 2 * P. Also, the interest rate is given as 5%, or 0.05 in decimal form. Therefore, the equation becomes:
2P = P * e^(0.05t)
Now, we can simplify the equation by dividing both sides by P:
2 = e^(0.05t)
To solve for t, we need to isolate the exponent. We can do this by taking the natural logarithm (ln) of both sides:
ln(2) = ln(e^(0.05t))
Using the property of logarithms, ln(e^(0.05t)) can be simplified to 0.05t * ln(e), but ln(e) is equal to 1:
ln(2) = 0.05t
Now, we can solve for t by dividing both sides by 0.05:
t = ln(2) / 0.05
Using a calculator, we find that ln(2) is approximately 0.693147:
t ≈ 0.693147 / 0.05
t ≈ 13.86 years
Therefore, it will take approximately 13.86 years to double your money at a 5% interest rate, compounded continuously.