Are elements of the range of a transcendental function themselves transcendental, excepting a "few" special cases?

Question

Let $f(x)$ be a transcendental function with $x\in\mathbb{C}$. Then are the values $f(x)$ themselves transcendental, except perhaps for a "few" exceptions?

For example, it is known that $f(x)=e^x$ is transcendental, and clearly $f(0)=e^0=1$ is algebraic, but what of all the other values? Are they transcendental? What about for other transcendental functions $f$ ?

EDIT: I suppose in the case of $f(x)=e^x$ we have things such as $x=\text{ln}(a),$ where $a$ can be anything we want. What if we only allow algebraic values $x$ ?

Answer 1

All numbers are transcendental except for a few special cases, i.e. all except countably many. As for $e^x$, notice that every positive number belongs to its range. Thus $5$, for example, is a member of the range.

Answer 2

There are countably many values of $x$ for which $f(x)$ is algebraic, and even rational. As an example, let $x = \ln{r}$ for any $r \in \mathbb{Q^+}$. In fact, continuous transcendental functions will generally have at least countably many values which are rational, by the intermediate value theorem.