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Math Help - log proof

  1. #1
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    log proof

    prove log base 2 of 3 is irrational
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by teramaries View Post
    prove log base 2 of 3 is irrational
     \log_23=\frac ab \implies 2^{a/b}=3\implies 2^a=3^b

    Now look at the prime factorization of both sides and see this forces  a=b=0 which is absurd in our case.
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  3. #3
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    Quote Originally Posted by teramaries View Post
    prove log base 2 of 3 is irrational
    Use proof by contradiction.

    Assume \log_2 3 = \frac{p}{q} where p and q are co-prime positive whole numbers

     \Rightarrow 3^q = 2^p.

    Now consider the last digits of 3^q and 2^p to get the required contradiction.
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  4. #4
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    A slight variation of previous proofs . . .


    Assume \log_23 = \frac{p}{q} where p and q are coprime positive integers.

    Then: . 2^{\frac{p}{q}} \:=\:3 \quad\Rightarrow\quad 2^p \:=\:3^q


    Since p is a positive integer, 2^p is even.

    Since q is a positive integer, 3^q is odd.

    Hence: . 2^p \:\ne\:3^q

    . . Q.E.D.
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