  Numbers are often used to define more complicated numbers. For example, two
integers are used to define a rational number and two reals are used to define
a complex number. It might be expected that an irrational power of an
irrational number would be an irrational number. Despite that, it is actually
possible for an irrational power of an irrational number to be a rational
number. For many generations, this circulated as a non-constructive proof by
contradiction in logic for discrete mathematics textbooks and college courses.
Since the 80s, a constructive proof circulated orally, such as
$\sqrt{2}^{log_\sqrt{2} 3}$ equals to 3. A written proof was published in 2008
by Lord. The first contribution of this paper is to prove that there is an
uncountable number of such pairs of irrational numbers such that the power of
one to the other is a rational number. Marshall and Tan answered the question
of whether there is a single irrational number $a$ such that $a^a$ is rational.
They proved that given $I$ = $(\frac{1}{e})^{(\frac{1}{e},\infty)}$, then every
rational number in $I$ is either of the form $a^a$ for an irrational $a$ or is
in the very thin set $\{1,4,27,256,...,n^n,...\}$. It seems a challenging task
to analytically solve the equation $x^x=y$ for any real $y$. To the best of our
knowledge, there is no work on finding $x$ from a given $y$. We proved that
$ln((ln(y))<x$ for $y>e$ and $x<ln(y)$, for $y> e^e$. Hence, the second
contribution of this paper is to estimate the real $x$ such that either one of
the equations $x^x=y$ or $x^{x^x}=y$ holds, for a given $y$.
