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Homework Help: Proving a number is irrational.

  1. Dec 20, 2011 #1
    1. The problem statement, all variables and given/known data
    Prove that [itex] log_2(3) [/itex] is irrational.
    3. The attempt at a solution

    This is also equivalent to [itex] 2^x=3 [/itex] from the definition of logs.
    Proof: For the sake of contradiction lets assume that x is rational and that their exists integers P and Q such that x=P/Q .
    so now we have [itex] 2^{\frac{P}{Q}}=3 [/itex]
    now I will take both sides to the Q power .
    so now we have [itex] 2^P=3^Q [/itex]
    since P and Q are integers, there is no possible way to have 2 raised to an integer to equal 3 raised to an integer, because 2^P will always be even and 3^Q will always be odd. so this is a contradiction and therefore x is irrational.
  2. jcsd
  3. Dec 20, 2011 #2
    Looks good :)
  4. Dec 20, 2011 #3
    sweet ok , I'm new to writing proofs so just want some confirmation.
  5. Dec 20, 2011 #4
    I can't imagine that you would lose points for this, but for the sake of pedantry you might want to point out that P and Q would have to both be positive integers. Just because 2^0=3^0 and 2^P, 3^Q aren't even and odd respectively when P and Q are negative.
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