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Logarithm: Compound interest problem

  1. Dec 27, 2012 #1
    1. The problem statement, all variables and given/known data

    How long does it take for a sum of money to double if it is invested at 8% compounded semi-annually?

    2. Relevant equations
    A = P(1+i)n

    A: Compounded amount
    P: Initial amount
    i: Interest rate
    n: Period

    3. The attempt at a solution
    A = P(1+i)n
    (2x) = (x)[1+(0.08)]2n (2n because it's compounded semi annually)
    2 = 1.082n (x cancels out)
    2n = log1.082
    n = (log1.082)/2
    n = 4.5032 per half a year

    Although the answer given at the back of the package is 8.836a.
    And I'm assuming "a" stands for annual. So I'm not sure where I went wrong.
    Any help is much appreciated! Thanks in advance!
     
  2. jcsd
  3. Dec 27, 2012 #2

    cepheid

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    Staff Emeritus
    Science Advisor
    Gold Member

    When you're compounding at some period other than annually, like say at intervals of 1/N years, the equation is:

    A = P(1+i/N)n

    and it's understood that n is the number of compounding periods, which means it represents the time interval in units of Nths of a year, rather than years.

    Edit: In other words, you apply one Nth of the interest rate, and you do this N times a year.

    In this case, we compound semi-annually, or every 1/2 year, so N = 2.

    So, we have

    log(A) = log(P) + nlog(1 + 0.08/2)

    log(2P) = log(P) + nlog(1.04)

    log(P) + log(2) - log(P) = nlog(1.04)

    n = log(2)/log(1.04)

    n = 17.6729876851

    So, the total time required is 17.673 HALF-YEARS (compounding periods). Divide that by 2 to get 8.836 years. I believe the 'a' stands for 'annum', which is Latin for 'year'.
     
  4. Dec 27, 2012 #3
    ohhhh okay thanks alot!
     
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