Question

log2 rational or irrational? justify your answer.

Answer 1

 Assume that log 2 is rational, that is,

                                         (1)

                where p, q are integers.

                Since log 1 = 0 and log 10 = 1,  0 < log 2 < 1  and therefore p < q.

                From (1),          

                                                

                                                   , where q – p is an integer greater than 0.

                Now, it can be seen that the L.H.S. is even and the R.H.S. is odd.

                Hence there is contradiction and  log 2  is irrational.