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Math Help - log

  1. #1
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    Feb 2006
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    log

    question1:3^x+2 = 5 ^x-1
    question2:ln(x+1)+ln(x-1)=ln8
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  2. #2
    MHF Contributor
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    Quote Originally Posted by rachael
    question1:3^x+2 = 5 ^x-1
    question2:ln(x+1)+ln(x-1)=ln8
    Here is one way.

    3^(x+2) = 5^(x-1)
    Take the common log, or to the base 10, of both sides,
    (x+2)log(3) = (x-1)log(5)
    Expand both sides,
    x*log(3) +2log(3) = x*log(5) -log(5)
    Isolate the x-terms,
    x*log(3) -x*log(5) = -log(5) -2log(3)
    Too much negatives. Divide both sides by -1,
    x*log(5) -x*log(3) = log(5) +2log(3)
    x[log(5) -log(3)] = log(5) +log(3^2)
    x[log(5/3)] = log(5*9)
    x*log(5/3) = log(45)
    x = [log(45)]/[log(5/3)] = 7.451980309 -------------answer.

    Check,
    3^(x+2) = 5^(x-1)
    3^(9.451980309) =? 5^(6.451980309)
    32,340.0513 =? 32,340.0513
    Yes, so, OK.

    -------------------------------
    ln(x+1) +ln(x-1) = ln(8)
    ln[(x+1)(x-1)] = ln(8)
    ln(x^2 -1) = ln(8)
    x^2 -1 = 8
    x^2 = 8 +1
    x^2 = 9
    x = +,-3 -------***

    Check x = 3,
    ln(3+1) +ln(3-1) =? ln(8)
    ln(4) +ln(2) =? ln(8)
    ln(4*2) =? ln(8)
    ln(8) =? ln(8)
    Yes, so, OK.

    Check x = -3,
    ln(-3+1) +ln(-3-1) =? ln(8)
    ln(-2) +ln(-4) =? .....
    Umm, there are no logarithm of negative numbers....
    So, no good. Reject x = -3.

    Therefore, x=3 ----------answer.

    Why no logarithm of negative number?
    See this example.
    If log(-a) = b
    Then -a = 10^b.
    No amount of b can result to -a.

    Another.
    ln(-a) = b
    -a = e^b
    No amount of b can result to -a.

    Another.
    ln(-a) = -b
    -a = e^(-b)
    -a = 1/(e^b)
    No amount of b can result to -a.
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  3. #3
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    thank you
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