Is ln sqrt(2) irrational? I know that the natural log of any positive algebraic number is transcendental, as a consequence of the Lindemann-Weierstrass theorem, but what about the natural log of the square root of two (which is irrational). Is this rational or irrational?

zapri4j

zapri4j

Answered question

2022-09-17

Is ln 2 irrational?
I know that the natural log of any positive algebraic number is transcendental, as a consequence of the Lindemann-Weierstrass theorem, but what about the natural log of the square root of two (which is irrational).
Is this rational or irrational?

Answer & Explanation

berzamauw

berzamauw

Beginner2022-09-18Added 9 answers

Not only is ln ( 2 ) irrational, but it's also transcendental!
Proof:
ln ( 2 ) = ln ( 2 1 / 2 ) = 1 2 ln ( 2 ) T
which is transcendental.
For reference, T is the set of transcendental numbers.
Makaila Simon

Makaila Simon

Beginner2022-09-19Added 2 answers

If you already know that the log of a positive algebraic number is transcendental, then all you need to realize is that 2 is a positive algebraic number. 2 is a root of x 2 2 = 0
Therefore, log ( 2 ) is transcendental log ( 2 ) is irrational.

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