themobius6s

2022-09-30

Logarithm of a transcendental number

Can anything be said about the nature of the number $\mathrm{log}y$ where $y$ is a transcendental number not of the form $y={e}^{x}$ or written trivially in that form using $x=\mathrm{log}w$ for some $w$ transcendental? Would the result always be transcendental? Just considering real numbers here.

Can anything be said about the nature of the number $\mathrm{log}y$ where $y$ is a transcendental number not of the form $y={e}^{x}$ or written trivially in that form using $x=\mathrm{log}w$ for some $w$ transcendental? Would the result always be transcendental? Just considering real numbers here.

Alannah Hanson

Beginner2022-10-01Added 11 answers

What can be said about $\mathrm{ln}y$ where $y$ is not of the form $y={e}^{x}$?

That such a value can never exist. Because only negative numbers aren't of the form $y={e}^{x}$ and the natural logs of negative numbers don't exist.

Is it always transcendental?

No, it is never transcendental because it never exists.

If $y$ is non-negative and it's not of the form $y={e}^{x};x$ not transcendental? Is that always transcendental?

Yes. If $y$ is not of the form $y={e}^{x};x$ not transcendental then $y$ is of the form $y={e}^{x};x$ is transcendental, so $\mathrm{ln}y=x$ is transcendental.

That such a value can never exist. Because only negative numbers aren't of the form $y={e}^{x}$ and the natural logs of negative numbers don't exist.

Is it always transcendental?

No, it is never transcendental because it never exists.

If $y$ is non-negative and it's not of the form $y={e}^{x};x$ not transcendental? Is that always transcendental?

Yes. If $y$ is not of the form $y={e}^{x};x$ not transcendental then $y$ is of the form $y={e}^{x};x$ is transcendental, so $\mathrm{ln}y=x$ is transcendental.

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