How to solve $ {z = x^y, x = y^z, y = z^x }$ for $ x, y $ and $ z? $
Is some sort of triple Lambert W to be introduced?
Done so far:
Taking logs,
$$ \log z = y \log x , \log x = z \log y , \log y = x \log z ; $$
Plug the third into second and that again into the first and simplify to get cyclically:
$$ x\, y \, z = x\, y \cdot log _y x = y\, z\cdot log _z y = z\, x\cdot log _x z = 1 ; $$
Then, how to proceed next?
Plots of $y= x^x$ and and $x \cdot y\, log_y x = 1 $ pass through (0,1) and (1,1).
So, is (x,y,z) = (1,1,1) the only solution?
My motivation for the post:
Solution of $ x^2 = 2^x $ has one more real root apart from inspection originated
solution x=2. Similarly there could be some other, may be complex roots at least.
Assuming all three variables are positive reals, let us rewrite the three equations as $\quad\sqrt[{\LARGE z}]z=y^y,$ $\sqrt[{\LARGE y}]y=x^x,\quad\sqrt[{\LARGE x}]x=z^z.~$ Then let us extract one expression for $z=z_1(x)$ from the last equation, and another one from the first two equations, $z=z(y)=z_2(x)$, since $y=y(x)$ from the second equation. Plotting them, we see that the two graphics intersect only for $x=1$.