Question

Is log 7 rational or irrational? Justify your answer

Answer 1

It'll be hard to prove, since it's not true: log_2 (7) is irrational.

Proof: If log_2 (7) is some rational number p/q, then 2^(p/q) = 7, so 2^p = 7^q. This implies the same number has two different factorizations into prime numbers, which is impossible. So by contradiction, log_2 (7) can't be rational.