To make things more complex, the same function can have transcendental/
algebraic number answers depending on the
domain. For example,
sin (1
radian) is transcendental, while
sin (1
degree) is algebraic.
Since e is transcendental, ln a (where a is not equal to 0 or 1 and is algebraic) is also transcendental.
A fundamental problem about transcendental numbers that still needs to be proved is whether a^b, where a is not equal to 0 or 1 and is
algebraic and b is
algebraic but not
rational, is
always transcendental.