# Proving a number is irrational.

1. Dec 20, 2011

### cragar

1. The problem statement, all variables and given/known data
Prove that $log_2(3)$ is irrational.
3. The attempt at a solution

This is also equivalent to $2^x=3$ 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 $2^{\frac{P}{Q}}=3$
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.

2. Dec 20, 2011

### JHamm

Looks good :)

3. Dec 20, 2011

### cragar

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

4. Dec 20, 2011

### 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.