Assume that log 2 is rational, that is,
where p, q are integers. Since log 1=0 and log10=1,0<log2<1 and p<q.
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.