Prove that log

istremage8o

istremage8o

Answered question

2022-05-29

Prove that log 2 3 is irrational
Seemingly simple homework assignment. Was never the best with logarithms, how would I go about proving?

Answer & Explanation

Louis Lawrence

Louis Lawrence

Beginner2022-05-30Added 10 answers

Let log 2 3 = p / q where p Z and q N (since surely log 2 3 > 0 you may directly assume that p N as well.)
Now it must hold
2 p = 3 q
But note that one side is even and the other one is odd!
Hence log 2 3 is not rational!
Kozankaht2

Kozankaht2

Beginner2022-05-31Added 2 answers

Assume, x := l o g 2 ( 3 ) = a b
As x > 0 , we can assume that a and b are positive integers.
Now
2 x = 3
leads to
2 a b = 3Therefore
2 a = 3 b
If a and b are positive, the left side is even, the right side odd. A clear contradiction.

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