1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook