Question

log2 rational or irrational? Justify your answer.

Answer 1

 Assuming that log 2 is a rational number. Then it can be expressed as  with  and  are positive integers (Why?).  Then, the equation is equivalent to . Raising both sides of the equation to , we have . This implies that .  Notice that this equation cannot hold (by the Fundamental Theorem of Arithmetic) because  is an integer that is not divisible by 5 for any , while  is divisible by 5. This means that log 2 cannot be expressed as  and is therefore irrational which is what we want to show.