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How do we know if Log(2)_3 is not equal to something like ((x^y)+a)

  1. Aug 11, 2012 #1
    How do we know if Log(2)_3 is not equal to something like ((x^y)+a) ,for rational a,x,y ?
     
    Last edited: Aug 11, 2012
  2. jcsd
  3. Aug 11, 2012 #2

    micromass

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  4. Aug 11, 2012 #3
    Re: Logarithms

    thanks, but how is
    equivalent to
    ?
     
  5. Aug 11, 2012 #4

    micromass

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    Re: Logarithms

    What is
    [tex]\alpha^{\log(\gamma)/\log(\alpha)}[/tex]
    ?
     
  6. Aug 11, 2012 #5
    Re: Logarithms

    you mean [tex]\gamma [/tex] ?
     
  7. Aug 11, 2012 #6

    micromass

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    Re: Logarithms

    Yes.

    So, IF [itex]\log(\gamma)/\log(\alpha)[/itex] were an algebraic nonrational number, then by applying Gelfond-Schneider we get ...
     
  8. Aug 11, 2012 #7
    Re: Logarithms

    I get it
     
  9. Aug 11, 2012 #8
    Re: Logarithms

    thanks
     
  10. Aug 11, 2012 #9

    micromass

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    Re: Logarithms

    Of course, to be able to apply Gelfond-Schneider to [itex]\log(2)/\log(3)[/itex], we must first prove that it's not rational...
     
  11. Aug 11, 2012 #10
    Re: Logarithms

    Which is easy, isnt it?

    since 2 ^a for integer a is never a power of 3...
    Right?
     
  12. Aug 11, 2012 #11

    micromass

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    Re: Logarithms

    Yeah, that's it (except for a=0 of course, but that's also not possible). I just wanted to point it out in case you missed it :smile:
     
  13. Aug 11, 2012 #12
    Re: Logarithms

    I actually did.
     
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