Suppose it’s rational. Then log₁₀ 5 = a/b, where a and b are positive integers. Then 10^a/b = 5 and 10^a = 5^b. But these two values can never be equal — any positive integral power of 10 will end in 0, and any such power of 5 will end in 5. So log₁₀ 5 cannot be rational.