Question

log 5 is irrational why ?

Answer 1

Suppose that log5(8)=mnlog5⁡(8)=mn, where m and n are integers and n is nonzero. Then 5mn=85mn=8, i.e. 5m=8n=23n5m=8n=23n.

Hence the LHS is purely a power of 5 and the RHS is a power of 2 only, which goes against the uniqueness of prime factorization.

This proves that the log value is irrational.