1. Limited time only! Sign up for a free 30min personal 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!

Are my proofs 'correct'?

  1. Aug 23, 2011 #1
    I can never seem to create proofs the way it is shown in every textbook I've seen. To be honest, I don't really know how to write the proofs correctly. I've seen sometimes my reasons are flawed and other times I go around aimlessly and get home after some unnecessary steps. So I would just like some feedback here:

    1. The problem statement, all variables and given/known data
    Prove that (log10 a)/(log10 b) is not rational if a and b are relatively prime and both are greater than 1.


    2. Relevant equations



    3. The attempt at a solution
    Suppose that (log10 a)/(log10 b) is rational, then they are expressed as fraction:
    (log10 a)/(log10 b) = p/q, where p and q are integers. q not 0. p not 0 because a is greater than 1.
    Let 10x = a
    and 10y = b
    x and y cannot be zero because a and b cannot be 1.
    Now if a and b are relatively prime, ie they have only 1 as their common factor then x and y too are relatively prime.

    Reason:
    Suppose that x and y share a common factor greater than 1, then:
    let x = kc
    and y = kz
    where k is the common factor, which is not negative or 0 because a and b are greater than 1 and c and z are also not 0 or negative.
    So then, 10kc = a
    and 10kz = b
    x = log10 a
    y = log10 b
    So, kc = log10 a
    and kz = log10 b
    But then, (10k)c = a
    and (10k)z = b
    And this contradicts the fact that a and b are relatively prime. So x and y are also relatively prime.

    So now we can write:
    x/y = p/q
    i.e, xq = py
    This cannot be true because x and y are relatively prime. This means x/y is not rational.
    And since x = log10 a
    and y = log10 b
    (log10 a)/(log10 b) is not rational.

    Thank-you
     
    Last edited: Aug 23, 2011
  2. jcsd
  3. Aug 23, 2011 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi Thinker8921! :smile:

    hmm … this is really complicated :redface:

    you're introducing x and y for no particular reason. :confused:

    Just do …
    … then write that as q(log10 a) = p(log10 b), and carry on from there. :smile:

    (or better still use logb a, if you know how to do that)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook