  We study algebraic and transcendental powers of positive real numbers,
including solutions of each of the equations $x^x=y$, $x^y=y^x$, $x^x=y^y$,
$x^y=y$, and $x^{x^y}=y$. Applications to values of the iterated exponential
functions are given. The main tools used are classical theorems of
Hermite-Lindemann and Gelfond-Schneider, together with solutions of exponential
Diophantine equations.
