Show that the logarithm in base 4+sqrt(5) of 6+sqrt(5) is irrational.The proof proceeds by contradiction. Assume the logarithm in base 4+sqrt(5) of 6+sqrt(5) equals p/q with p,q positive integers. Then
(4+sqrt(5))^p=(6+sqrt(5))^q (*)By taking conjugates,
(4-sqrt(5))^p=(6-sqrt(5))^q (**)Multiplying (*) and (**) term by term we get
11^p=31^q.The fundamental theorem of arithmetic tells us this cannot happen.