Prove that log5(2) is irrational. (Hint: Use the unique factorisation of integers theorem....

Prove that log5(2) is irrational. (Hint: Use the unique factorisation of integers theorem.)

Unique Factorization of Integers Theorem (Fundamental Theorem of Arithmetic)

Given any integer n > 1, there exist a positive integer k, distinct prime numbers p1, p2, …, pk , and positive integers e1, e2, …, ek such that

and any other expression for n as a product of prime numbers is identical to this except, perhaps, for the order in which the factors are written.