Let us assume log 2 as rational,that is
  log 2=p/q -------1
where p,q are integers.
Since log 1 =0 and log 10=1,0<log 2<1 and therefore p<q.
from 1,   2=10powerp/q
              2powerq=(2*5)power p
           2power q-p= 5 power p, where q-p is an interger greater than 0.
Now it can be seen that L.H.S is even and the RHS is odd.Hence there is contradiction and therefore log2 is irrational.