Question

is log2 rational or irrational?justify your answer

Answer 1

Assuming that log 2 is a rational number. Then it can be expressed as A\B with A and B are positive integers (Why?).  Then, the equation is equivalent to 2=10A\B. Raising both sides of the equation to B, we have 2B(where B is power)=10A(where A is power). This implies that 2B=5A 2A(where A and B are power).  Notice that this equation cannot hold (by the Fundamental Theorem of Arithmetic) because 2B(where B is power) is an integer that is not divisible by 5 for any B, while 2A 5A(where A and B are power) is divisible by 5. This means that log 2 cannot be expressed as A\B and is therefore irrational which is what we want to show.