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Geometric Sequences and Logarithms

  1. Sep 11, 2003 #1
    I'm having trouble with these type of probles (where a negative log comes up):

    (All of this is solving without sigma notation)

    Find the number of terms in these geometric sequences and the sum of the numbers.

    11, -22, 44,....,704

    I know that a1 = 11, r = -2, and an = 704, so I did:

    704 = 11(-2)^n-1 so,

    64 = (-2)^n-1

    log 64 = (log -2) n-1
    n should = 7, but when I found the log of 64 divided by the log of -2, I got .2785219413 - 1.2...

    Why is there a minus sign and what am I doing wrong? Please help. Thanks!
    Last edited: Sep 11, 2003
  2. jcsd
  3. Sep 11, 2003 #2


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    Staff Emeritus

    Ignore the fact that the ratio is negative, since it will cause problems with the logarithm. (You can't take the log of a negative number)

    the equation should start out as

    704 = 11*2^x

    so basically you get

    11 * 2 * 2 * 2 * ..... = 704

    you can simpliffy the equation by dividing both sides of the equation by 11

    2^x = 64

    x = ln(64)/ln(2) = 6

    Then you have to add 1 to account for the first term

    therefore the number of terms is 7
  4. Sep 11, 2003 #3


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    Staff Emeritus
    Science Advisor
    Gold Member

    You don't just ignore the negative sign; you take the absolute value of both sides.

    (and because it's a nonreversible operation, you mave to check your answer to make sure it works)

    And, incidentally, you can take the logarithm of a negative number, but it does require you to take a step up to complex analysis... the logarithm of a negative number is a (nonunique) complex number, and it takes a lot of care to make sure you are doing everything right.

    When you computed log 64 / log -2, the result is thus a complex number and your calculator reported the principal value of this expression. If you had scrolled your display to the right, you would have seen 'i' in the answer
  5. Sep 11, 2003 #4
    THANK YOU!!!!!!!!
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