Is it correct that the natural logarithm function maps algebraic numbers to transcendental and transcendental numbers to algebraic, other than 1? Of course, over the domain natural log is defined i.e. $(0,\infty)$?
i.e.
$$\ln:A^+ \rightarrow T \hspace{5 mm};\hspace{5 mm} \ln:T^+ \rightarrow A,$$
and$$\ln:A^+ \nrightarrow A \hspace{5 mm};\hspace{5 mm} \ln:T^+ \nrightarrow T.$$
where;
$A^+$ is the set of positive algebraic numbers except $1$.
$T^+$ is the set of positive transcendental numbers.
$T$ is the set of transcendental numbers.
$A$ is the set of algebraic numbers.
It is impossible for cardinality reasons alone, since $\ln$ is injective, so it cannot map an uncountable set into a countable one.
It is true that $\ln$ maps positive algebraic numbers (except 1) to transcendental numbers, by the Lindemann–Weierstrass theorem. Every algebraic real number except $0$ is the logarithm of a positive transcendental number, but so are most transcendental numbers. For a concrete example, $e^\pi$ is a transcendental number whose logarithm is transcendental.