Proving a number is irrational.

[cragar]

Homework Statement

Prove that log_2(3) is irrational.

The Attempt at a Solution

This is also equivalent to 2^x=3 from the definition of logs.
Proof: For the sake of contradiction let's assume that x is rational and that their exists integers P and Q such that x=P/Q .
so now we have 2^{\frac{P}{Q}}=3
now I will take both sides to the Q power .
so now we have 2^P=3^Q
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.

 

[JHamm]

Looks good :)

 

[cragar]

sweet ok , I'm new to writing proofs so just want some confirmation.

 

[Stimpon]

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.