Logarithm of a transcendental number

Question

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

Answer 1

What can be said about $\ln y$ 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 $\ln y = x$ is transcendental.

Answer 2

Every positive number is of the form $y=e^x$ for some $x$, and negative numbers do not have logarithms.

Answer 3

No. $e$ is transcendental and $\log e = 1$.

And every positive real number $y$ can be written as $e^x$ for some $x$ (specifically $x=\log y$).