Let log52=y Let us assume log52 is a rational number. log52=p/q where p and q are co primes. 5(
p
q
)=2=>5p=2q 5p=5×5×5×5×5×5×5...........p times 2p=2×2×2×2×2×2×2..........q times No value of p and q can satisfy the equation. Hence y is an irrational number.