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Proving a log identity

  1. Mar 20, 2013 #1

    K29

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    1. The problem statement, all variables and given/known data

    [itex]a^{log_{b}(c)}=c^{log_{b}(a)}[/itex]


    3. The attempt at a solution
    Take [itex]log_{a}[/itex] of both sides:
    [itex]log_{a}(a^{log_{b}(c)})=log_{a}(c^{log_{b}(a)})[/itex]

    gives:
    [itex]log_{b}c=log_{b}alog_{a}c[/itex]

    Looks like one more step for the RHS. I sort of see that the RHS should become [itex]log_{b}c[/itex] and then we'll be done. But standard log laws don't seem to help me make this step. Please help with this step/explain why this is so?
     
  2. jcsd
  3. Mar 20, 2013 #2

    SammyS

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    Using the commutative law of multiplication, the RHS is [itex]\displaystyle \ (\log_{a\,}c)\,(\log_{b\,}a)\quad\to\quad\log_{b\,}\left(a^{\log_{a\,}c}\right)[/itex]
     
  4. Mar 20, 2013 #3

    K29

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    Got it. Thanks a bunch
     
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