Is there any correlation between $A^B$ and $B^A$?

Question

So, we all know that exponentiation is non-commutative, but is there any relationship despite exponentiation's not being commutative? Is there any correlation between $A^B$ and $B^A$ where $A$ and $B$ are real numbers?

Answer 1

There is no function $f$ such that $f(A^B)=B^A$ (for all real numbers $A,B$ such that $A^B$ and $B^A$ are defined). If there were such a function $f$, then for example we would have $$36=6^2=f(2^6)=f(4^3)=3^4=81,$$ which is clearly not the case.

Now, for any positive $A\neq 1$, there is a function $f_A$ such that $f_A(A^B)=B^A$ for any $B$. In particular, $$f_A(x):=\left(\frac{\log x}{\log A}\right)^A.$$ Also, for any positive $B$, there is a function $g_B$ such that $g_B(A^B)=B^A$ for any $A$. In particular, $$g_B(x):=B^{\frac{\log x}{B}}.$$