 # 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 2022-09-17 Answered
Is $\mathrm{ln}\sqrt{2}$ 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?
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Not only is $\mathrm{ln}\left(\sqrt{2}\right)$ irrational, but it's also transcendental!
Proof:
$\mathrm{ln}\left(\sqrt{2}\right)=\mathrm{ln}\left({2}^{1/2}\right)=\frac{1}{2}\underset{\in \mathbb{T}}{\underset{⏟}{\mathrm{ln}\left(2\right)}}$
which is transcendental. $◻$
For reference, $\mathbb{T}$ is the set of transcendental numbers.

We have step-by-step solutions for your answer! Makaila Simon
If you already know that the log of a positive algebraic number is transcendental, then all you need to realize is that $\sqrt{2}$ is a positive algebraic number. $\sqrt{2}$ is a root of ${x}^{2}-2=0$
Therefore, $\mathrm{log}\left(\sqrt{2}\right)$ is transcendental $\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}$$\mathrm{log}\left(\sqrt{2}\right)$ is irrational.

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