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Math Help - Prove that z=log base p (qr) is an irrational number

  1. #1
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    Prove that z=log base p (qr) is an irrational number

    Hello everyone,
    I'd like to prove that, providing p q and r are distinct but arbitrary prime numbers that z=logp(qr) is an irrational number. I know it's supposed to be a proof by contradiction but I'm having tons of problems getting through it. Could anyone help?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by JTL4869 View Post
    Hello everyone,
    I'd like to prove that, providing p q and r are distinct but arbitrary prime numbers that z=logp(qr) is an irrational number. I know it's supposed to be a proof by contradiction but I'm having tons of problems getting through it. Could anyone help?
    Suppose otherwise, then there exist (coprime) integers $$a and $$b such that:

    \dfrac{a}{b}=\log_p(qr)

    or:

    (p)^a=(qr)^b

    and you should be able to get your contradiction from the fundamental theorem of arithmetic

    CB
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  3. #3
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    Thank you! I'm assuming the contradiction is from the uniqueness part of the FTA?
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by JTL4869 View Post
    Thank you! I'm assuming the contradiction is from the uniqueness part of the FTA?
    Yes.

    CB
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