Question that can not be solve analytically . You can know that the solution of this non-linear simultaneous equations is y=2 and x=3; but the question is : How can mathematically ( algebraically ) find this.
\begin{array}{lcl} x^y &amp; = &amp; 9 \\ y^x &amp; = &amp; 8 \end{array}
This is equivalent to solving 

$$ x^{a^x} = b $$

which, I think, has no analytic solution.
Ln both equations to get:

$\ln x^y = \ln 9$ and $\ln y^x = \ln 8$

They then become $y \ln x = \ln 9$ and $x \ln y = \ln 8$

Solving for $x$ and $y$ you get 

$y = \frac{\ln 9}{\ln x}$ and $x = \frac{\ln 8}{\ln y}$

Substituting $x = \frac{\ln 8}{\ln y}$ into the first equation you get

$y = \frac{\ln 9}{\ln(\ln 8/\ln y)}$ 

Doing the same for $x$, you get 

$x = \frac{\ln 8}{\ln(\ln 9/\ln x)}$

...?