If $a\in\mathbb{R}\setminus\left\{0,1\right\}$ is an algebraic number, can $\ln\left(a\right)$ ever be a Liouville number?
This is not a homework question, nor do I know much about the innards of proving these kinds of things. I am just very interested in transcendental numbers.
On
The answer to your question is no. This is the case because the exponential of each Liouville number is transcendental. See my paper here.It is open access.
Almost certainly it can't, but I would be surprised if this were provable in the current state of the art, even in the case where $a$ is rational.