SOLUTION: How long will it take to earn your first million dollars if you annually invest $1000 into an IRA at 10% interest compounded monthly. This answer should be calculated using logar

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Question 967716: How long will it take to earn your first million dollars if you annually invest $1000 into an IRA at 10% interest compounded monthly.
This answer should be calculated using logarithms so that the answer is exact.
Found 2 solutions by MathLover1, MathTherapy:
Answer by MathLover1(20855) About Me  (Show Source):
You can put this solution on YOUR website!

To calculate this value exactly, a fairly simple formula can be utilized.
A=P%281%2B%28r%2Fn%29%29%5E%28n%2At%29
P=principal (1000)
r=annual nominal interest rate (.10)
n=number of compounding during year (12, because of 12 months)
t=number of years (unknown)
A=amount after t years (first 1000000)
This formula yields results when we have the time known, but in this case we need to isolate "t" for the time necessary for the principal to turn into $1000000. So we need to rearrange the formula into a fairly complex logarithmic formula:
log%28A%29=log%28P%281%2B%28r%2Fn%29%29%5E%28n%2At%29%29
log%28A%29=log%28P%29%2Blog%28%281%2B%28r%2Fn%29%29%5E%28n%2At%29%29%29
log%28A%29-log%28P%29=log%28%281%2B%28r%2Fn%29%29%5E%28n%2At%29%29%29
log%28A%2FP%29=%28n%2At%29log%28%281%2B%28r%2Fn%29%29%29
log%28A%2FP%29%2Flog%28%281%2B%28r%2Fn%29%29%29=%28n%2At%29
t=log%28A%2FP%29%2F%28%28n%2Alog%281%2B%28r%2Fn%29%29%29%29
Then substitute the numbers into the equation -
t=log%281000000%2F1000%29%2F%28%2812%2Alog%281%2B%280.10%2F12%29%29%29%29
t=log%281000%29%2F%2812%2Alog%281.008333333333333%29%29
t=69.36
Finally we got the answer - it'll take about 69 years and a four months!!!

Answer by MathTherapy(10858) About Me  (Show Source):
You can put this solution on YOUR website!
How long will it take to earn your first million dollars if you annually invest $1000 into an IRA at 10% interest compounded monthly.
This answer should be calculated using logarithms so that the answer is exact.
Use Future Value of Ordinary Annuity formula: FV%5Boa%5D+=+PMT+%2A+%28%28%281%2Bi%2Fm%29%5E%28mt%29-1%29%2F%28i%2Fm%29%29%29

1000000+=+1000+%2A+%28%28%281+%2B+.1%2F12%29%5E%2812t%29+-+1%29%2F%28.1%2F12%29%29%29

1000000%2F1000+=+%28%28%281+%2B+.1%2F12%29%5E%2812t%29+-+1%29%2F%28.1%2F12%29%29%29



1000+=+%28%28%28%2812+%2B+.1%29%2F12%29%5E%2812t%29+-+1%29%2F%28.1%2F12%29%29%29%29

1000+=+%28%28%2812.1%2F12%29%5E%2812t%29+-+1%29%2F%28.1%2F12%29%29%29

%2812.1%2F12%29%5E%2812t%29+-+1+=+100%2F12 ------ Cross-multiplying

%2812.1%2F12%29%5E%2812t%29+-+1+=+25%2F3

%2812.1%2F12%29%5E%2812t%29+=+25%2F3+%2B+1 

%2812.1%2F12%29%5E%2812t%29+=+25%2F3+%2B+3%2F3 

%2812.1%2F12%29%5E%2812t%29+=+28%2F3 ---------- Exponential form

log+%2812.1%2F12%2C+%2828%2F3%29%29+=+12t -------- Logarithmic form

log+%2828%2F3%29%2Flog+%2812.1%2F12%29+=+12t ---------- Applying change of base

%28log+%2828%2F3%29%29%2F%28log+%2812.1%2F12%29%29%22%F7%2212+=+t

highlight_green%28t+=+%28log+%2828%2F3%29%29%2F%2812+%2A+%28log+%2812.1%2F12%29%29%29%29 years ------ EXACT TIME

highlight_green%28t+=+22.42885983%29 years ------------ APPROXIMATE TIME