Question 1140028
f = p * (1 + 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.


that's the formula for discrete compounding.


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


f is the future value.
p is the present value.
e is the scientific constant equal to 2.718281828..... 
it is shown as the e ^ x key on your calculator.
r is the interest rate per time period.
n is the number of time periods.


their balance will be the same when the future value of their respective accounts are equal to each other.


this occurs when p * (1 + r) ^ n is equal to p * e ^ (r * n)


given your inputs, the formulas become:


4000 * (1 + .05) ^ n = 3750 * e ^ (.0495 * n)


divide both sides of this equation by 3750 to get:


4000 / 3750 * (1 + .05) ^ n = e ^ (.0495 * n)


divide both sides of this equation by (1 + .05) ^ n to get:


4000 / 3750 = (e ^ (.0495 * n) / ((1 + .05) ^ n)


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


ln(4000 / 3750) = ln((e ^ (.0495 * n) / ((1 + .05) ^ n))


since ln(a/b) is equal to ln(a) - ln(b), your equation becomes:


ln(4000 / 3750) = ln(e ^ (.0495 * n)) - ln((1 + .05) ^ n)


since ln(a^b) = b * ln(a), your equation becomes:


ln(4000 / 3750) = .0495 * n * ln(e) - n * ln(1 + .05)


since ln(e) is equal to 1, your equation becomes:


ln(4000 / 3750) = .0495 * n - n * ln(1 + .05)


factor out the n to get:


ln(4000 / 3750) = n * (.0495 - ln(1 + .05))


solve for n to get:


n = ln(4000 / 3750) / (.0495 - ln(1 + 05) = 90.92034856.


the balance in both accounts will be equal in 90.92034856 years.


4000 * (1 + .05) ^ 90.92034856 = 337,752.4038


3750 * e ^ (.0495 * 90.92034856) = 3347,752.4038


your solution is that the account balances will be equal in 90.92034856 years.


the steps involved might be easier to see in my hand drawn worksheet shown below.


<img src = "http://theo.x10hosting.com/2019/042801.jpg" alt="$$$" >


the transition from step 3 to step 4 takes advantages of the fact that ln(a/b) = ln(a) - ln(b).


the transition from step 4 to step 5 takes advantage of the fact that ln(a^b) = b * ln(a) and also takes advantage of the fact that ln(e) = 1.


that allows ln(e^(.0495*n)) to become equal to .0495 * n * ln(e) which then becomes equal to .0495 * n.


that also allows ln(1.05 ^ n) to become equal to n * ln(1.05).


step 7 factors out the n and then divides both sides of the equation by (.0495 - ln(1.05).


step 8 shows the result.


note that (1 + .05) is the same as 1.05.


showing 1.05 as (1 + .05) is done to reinforce the concept that it comes from the general expression of (1 + r).