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Math Help - Another Logarithm Equation: Not sure how to solve for x properly.

  1. #1
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    Another Logarithm Equation: Not sure how to solve for x properly.

    Hi again,

    I am stuck on another logarithmic equation:

    log3 x - 2 = log3 (x - 8)

    Now, I assume log3 cancels out, which leaves:

    x - 2 = x - 8

    But from there, I am unsure how you would get x by itself, because if you try and transfer x to just one side, it cancels out because you have x and -x. I think I am on the wrong track though, so all guidance will be much appreciated once again.

    Thanks,

    Nathaniel
    Last edited by mr fantastic; September 21st 2010 at 01:31 AM. Reason: Deleted words irrlevant to the purpose of the post title.
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  2. #2
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    Quote Originally Posted by BinaryBoy View Post
    Hi again,

    I am stuck on another logarithmic equation:

    log3 x - 2 = log3 (x - 8)

    Now, I assume log3 cancels out, which leaves:

    x - 2 = x - 8 Mr F says: It should be clear that this is nonsense as it implies -2 = -8.

    But from there, I am unsure how you would get x by itself, because if you try and transfer x to just one side, it cancels out because you have x and -x. I think I am on the wrong track though, so all guidance will be much appreciated once again.

    Thanks,

    Nathaniel
    The equation is possibly meant to be \displaystyle \log_3 (x) - 2 = \log_3 (x - 8) \Rightarrow \log_3 \left( \frac{x}{x - 8} \right) = 2 (and your job is to fill in the missing steps in getting this).

    Now solve for x.
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  3. #3
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    If the problem was, indeed, ln_3(x- 2)= ln_3(x- 8), then what you did is exactly correct- ln(x) is a "one-to-one function" so it must be true that x- 2= x- 8. But no matter what x is, that is the same as -2= -8 which is not true. NO value of x makes that equation true.

    But,as mr fantastic said, your lack of parentheses makes this problem ambiguous. It might be ln_3(x- 2)= ln_3(x- 8), ln_3(x)- 2= ln_3(x- 8), ln_3(x- 2)= ln_3(x)- 8, or ln_3(x)- 2= ln_3(x)- 8. The first and last of those, which would be the most reasonable interpretations, have no solution!
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  4. #4
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    Thanks for the help Mr Fantastic and HallsofIvy. I have solved for x and checked my answer and the equation is indeed of the form: log3 (x) -2 = logx (x - 8).

    So, what I did was:

    log3(x) = logx (x - 8) + 2

    Then I moved the logx(x - 8) to the other side by dividing both sides. So I get:

    log3(x / x - 8) = 2

    This turns into:

    log3( x / x - 8) = (3 to the power of 2)
    = 9

    Then I multiplied the x - 8 back over:

    log3(x) = 9(x - 8)

    log3(x) = 9x - 72

    Multiply the x over as well:

    0 = 8x - 72

    Move the 72 to the other side

    8x = 72

    Divide by 8 to yield:

    x = 9.

    Cool! Many thanks for the help!

    Nathaniel
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  5. #5
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    You should, of course, check that answer in the original problem:
    log_3(x)- 2= log_3(x- 8)

    Setting x= 9, log_3(9)= log_3(3^2)= 2 and log_3(9- 8)= log_3(1)= log_3(3^0)= 0 so the equation becomes 2- 2= 0 which is, in fact, correct.
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