Solve 3 exponential equations $z^x=x$, $z^y=y$, $y^y=x$ to get $x$, $y$, $z$.

Question

The main question is :

$z^x=x$, $z^y=y$, $y^y=x$

Find $z$, $y$, $x$.

My method :

I first attempted to get two equation for the unknowns $x$ and $y$.

We can happily write :

$z=x^{1/x}$ and $z=y^{1/y}$

Thus we get, $x^{1/x}=y^{1/y}$ Which is, $x^y=y^x$.

I can't go any farther than this. Please help me.

Answer 1

$$z^{y}=y$$ $$(z^y)^y=y^y$$ $$z^{y^2}=x=z^x$$ therefore $$x=y^2$$ on the other hand $$y^y=x=y^2\implies y=2$$ thus $$x=4\quad,\quad z=\sqrt{2}$$

Answer 2

One solution just by looking at it is $z=x=y=1$.

Answer 3

And to be different, the third solution is $x=y=z=-1$, by old-fashioned trial and error.