# Proving a number is irrational.

## 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 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.