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Math Help - Why is -ln8 equal ln1/8?

  1. #1
    Junior Member BugzLooney's Avatar
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    Why is -ln8 equal ln1/8?

    I solved this problem and couldn't get the last step but apparently it went like this. First e^-x=8
    Then ln(8)=-x
    So x=-ln(8)

    Apparently the answer is then x=ln(1/8)

    I just don't know why the negative ln8 is equal to 1/8.
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  2. #2
    Master Of Puppets
    pickslides's Avatar
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    x=-\ln(8) \implies x=\ln(8)^{-1} \implies x=\ln\left(\frac{1}{8}\right)

    All good?
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  3. #3
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    Quote Originally Posted by BugzLooney View Post
    I solved this problem and couldn't get the last step but apparently it went like this. First e^-x=8
    Then ln(8)=-x
    So x=-ln(8)

    Apparently the answer is then x=ln(1/8)

    I just don't know why the negative ln8 is equal to 1/8.
    That last line is not true. You need to use the following facts:

    \ln{1} = 0

    \ln{a} - \ln{b} = \ln{\left(\frac{a}{b}\right)}

    Then

    - \ln{8} = 0 - \ln{8} = \ln{1} - \ln{8} = \ln{\left(\frac{1}{8}\right)}


    edit: oops, beaten to it! My Latex skills are too slow.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by BugzLooney View Post
    I solved this problem and couldn't get the last step but apparently it went like this. First e^-x=8
    Then ln(8)=-x
    So x=-ln(8)

    Apparently the answer is then x=ln(1/8)

    I just don't know why the negative ln8 is equal to 1/8.
    The technical reason...is well technical. But think about it like this, \ln(x) is just the solution to e^y=x. So let \alpha=\ln\left(\frac{1}{8}\right) then e^{\alpha}=\frac{1}{8} or equivalently 8=\frac{1}{e^{\alpha}}=e^{-\alpha} so that \ln(8)=-\alpha\implies -\ln(8)=\alpha=\ln\left(\tfrac{1}{8}\right)
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  5. #5
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    e^(i*pi)'s Avatar
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    Of course neither -ln(8) nor ln\left(\frac{1}{8}\right) are fully simplified

    -3ln(2) = 3ln\left(\frac{1}{2}\right)
    Last edited by e^(i*pi); January 11th 2010 at 01:18 PM. Reason: that's what I get for being a smart-arse >.<
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  6. #6
    Super Member bigwave's Avatar
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    why would you say that is simplified??
    it doesn't look any like any reduced form?

    just being cranky.
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  7. #7
    Junior Member BugzLooney's Avatar
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    Woah how does ln(1/8) simplify to 3ln(2) and so forth?
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  8. #8
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by BugzLooney View Post
    Woah how does ln(1/8) simplify to 3ln(2) and so forth?
    \ln\left(ab\right)=\ln(a)+\ln(b) and \ln\left(a^n\right)=n\ln(a)
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  9. #9
    Junior Member BugzLooney's Avatar
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    I understand everything except for the last post as it relates to mine.

    ln(1/8) simplifies to 3ln(2) by that rule? I just don't see it. Though the explanations have been great thus far.
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  10. #10
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by BugzLooney View Post
    I understand everything except for the last post as it relates to mine.

    ln(1/8) simplifies to 3ln(2) by that rule? I just don't see it. Though the explanations have been great thus far.
    \ln(8)=\ln\left(2^3\right)=3\ln(2)
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  11. #11
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    Quote Originally Posted by BugzLooney View Post
    I solved this problem and couldn't get the last step but apparently it went like this. First e^-x=8
    Then ln(8)=-x
    So x=-ln(8)

    Apparently the answer is then x=ln(1/8)

    I just don't know why the negative ln8 is equal to 1/8.
    Since e^{-x}= 8, e^x= \frac{1}{8}.
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