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Math Help - word problem using logarithms

  1. #1
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    word problem using logarithms

    can someone plese help me with this question, how to set it up and solve it. thanks!

    a savings bond offers intrest at a rate of 6.6%, compounded semi-annually. suppose that a $500 bond is purchased. how long will it take for the investment to double in value.?
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  2. #2
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    If you're familiar with the basic 'single investment, compounding forward' set-up...

     500\left(1 \ + \ \frac{0.066}{2}\right)^n \ = \ F

    ...where F is the amount the initial investment will grow to after n periods, the solution to your question starts with that basic layout:

     500\left(1 \ + \ \frac{0.066}{2}\right)^n \ = \ 1,000

    which immediately reduces to

     \left(1 \ + \ \frac{0.066}{2}\right)^n \ = \ 2

    Now solve for n. Start by taking the logs of both sides, and remember that  \text{ln}(a^b) \ = \ b\cdot \text{ln}(a) . Finally, remember that in this set-up, n represents the number of semi-annual periods it'll take for the investment to double...so you might wanna re-express it as years in your final answer.
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  3. #3
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    Please do not forget the remarkably reliable "Rule of 72s". For reasonable interest rates, the precision is almost always more than expected.

    Since 6.6% is used semi-annually, we'll use 3.3% and calculate semi-annual periods.

    72 / 3.3 = 21.818, just a little too high.
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