Proof that log

dream13rxs

dream13rxs

Answered question

2022-07-04

Proof that log 2 3 + log 5 2 is irrational number

Answer & Explanation

Feriheashrz

Feriheashrz

Beginner2022-07-05Added 8 answers

2 l o g 2 3 .2 l o g 5 2 = 2 p q (where p and q are natural numbers as the number is graeter than 0).

(and a l o g b c = c l o g b a )

therefore 3 q . 2 q l o g 5 2 = 2 p (we know that 3 q and 2 p are natural number so we now just need to show 2 q l o g 5 2 is always irrational for any natural 'q').

2 q l o g 5 2 = m n (we know m n is not pf the form 2 k where k is a natural number becaise q l o g 5 2 cannot be natural)

therefore q = l o g 2 5. l o g 2 ( m n ) (where q is a natural number,and if l o g 2 m n becomes rational then l o g 2 5 being irrational q becomes irrational which is not possible.)

l o g 2 5 = a , l o g 2 m n = b ,where a and b are both irrational.I dont know how to proceed further , irrational × irrational can be rational.

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