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Logarithm Equation Problem

  1. Jan 18, 2009 #1
    1. The problem statement, all variables and given/known data
    Find x in the following equation

    (ln 64 / 2) + ln x = ln 18 - ln x

    The answer should be x = 3/2

    2. Relevant equations
    Logarithm laws

    ln m - ln n = ln (m/n)

    3. The attempt at a solution
    I tried two methods:

    (ln 64 / 2) + ln x = ln 18 - ln x
    ln 32 + ln x = ln 18 - ln x
    ln 32 - ln 18 = - ln x - ln x
    ln (32/18) = -2 ln x
    ln (32/18) = ln x-2
    (32/18) = x-2
    (32/18)1/-2 = x
    0.75 = 3/4 = x

    Wrong answer

    (ln 64 / 2) + ln x = ln 18 - ln x
    ln 64 + ln x = 2 (ln 18 - ln x)
    ln 64 + ln x = 2 ln 18 - 2 ln x
    ln 64 + ln x = ln 182 - ln x2
    ln 64 - ln 182 = - ln x - ln x2
    ln 64 - ln 324 = - ln x - 2 ln x
    ln 64 - ln 324 = - 3 ln x
    ln 64 - ln 324 = ln x-3
    ln (64/324) = ln x-3
    (64/324) = x-3
    (64/324)1/-3 = x
    1.717 = x

    Wrong answer again.

    The first steps seem to be more right to me though but I can't find what I did incorrect..

    Thank you in advanced. :smile:
     
  2. jcsd
  3. Jan 18, 2009 #2
    You've got the right idea but you are either messing your rules or not distributing properly.

    For the first one you did (ln64)/2 is eqaul to ln(32) which is not true

    The second one is more correct but when you distribute that 2 you forgot to multiply by lnx by 2.

    Your second line should look like this:
    ln 64 + 2(ln x) = 2 (ln 18 - ln x)

    you should be able to solve it from there
     
  4. Jan 18, 2009 #3

    Dick

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    Science Advisor
    Homework Helper

    I think you are interpreting the first term wrong. Instead of ln(64/2) do it with ln(64)/2. That will give you x=3/2.
     
  5. Jan 18, 2009 #4
    Thank you, I have resolved my problem with the following:

    ln 64 + 2 (ln x) = 2 (ln 18 - ln x)
    ln 64 + 2 ln x = 2 ln 18 - 2 ln x
    ln 64 - 2 ln 18 = -2 ln x - 2 ln x
    ln 64 - 2 ln 18 = -4 ln x
    ln 64 - ln 324 = ln x-4
    ln (64/324) = ln x-4
    (64/324) = x-4
    (64/324)1/-4 = x
    1.5 = x

    Thanks again, this is such a helpful resource.
     
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