I was wondering, could one define a two variable function such that
$$T \begin{pmatrix} a \\ b\\ \end{pmatrix} ^{T \begin{pmatrix} c \\ d\\ \end{pmatrix}}=T \begin{pmatrix} a^c \\ b^d\\ \end{pmatrix}$$
$∀ a,b,c,d$?
There are only a couple of things I have been able to figure out. Assuming that one could indeed define a function as the latter, then $T \begin{pmatrix} 1 \\ 1\\ \end{pmatrix}=1$, since $T \begin{pmatrix} a \\ b\\ \end{pmatrix} ^{T \begin{pmatrix} 1 \\ 1\\ \end{pmatrix}}=T \begin{pmatrix} a^1 \\ b^1\\ \end{pmatrix}$.
I also figured out that by assigning a value to a pairwise of variables in $T$, for example $$T\begin{pmatrix} 8 \\ 25\\ \end{pmatrix}=e$$, then there are other (infinitely many) values of T that become defined, for instance $$T\begin{pmatrix} 8^8 \\ 25^{25}\\ \end{pmatrix}=e^e$$
Any help/thoughts would be truly appreciated!
$T(x,y) = 1$ for all $x,y$ will do the trick. Note, however, that $a^c$ and $b^d$ might not always be defined if you are allowing all real numbers.