Suppose that log27 is rational number. ∴ Let log27=
p
q
, where p, q ∈ N. ⇒7=2
p
q
⇒7q=2p. Obviously (1) is false, as the L.H.S. is odd while the R.H.S. is even and the two cannot be equal. Thus,log27 is not a rational number. Also log27 is not an integer and so it cannot be a prime number. Hence, log27 is an irrational number.