Is $\ln\sqrt{2}$ irrational?

Question

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 1

Not only is $\ln(\sqrt{2})$ irrational, but it's also transcendental!

Proof: $$\Large \ln(\sqrt{2})=\ln(2^{1/2})=\frac{1}{2} \underbrace{\ln(2)}_{\in \mathbb{T}}$$ which is transcendental. $\square$

To see why the product of a transcendental number and a non-zero algebraic number is transcendental, see this .

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For reference, $\mathbb{T}$ is the set of transcendental numbers.

Answer 2

IRRATIONAL

because $\sqrt{2} = 2^{1/2}$ and hence $\ln(\sqrt{2} )= 1/2 \ln(2)=$ an irrational number

Answer 3

If you already know that the log of a positive algebraic number is transcendental, then all you need to realize is that $\sqrt2$ is a positive algebraic number. $\sqrt2$ is a root of $x^2-2=0$.

Therefore, $\log(\sqrt2)$ is transcendental $\implies$ $\log(\sqrt2)$ is irrational.

Answer 4

Think that $\log\sqrt2=\frac12\log2$ !