Hello guys,

I was doing some work with functions, and I came across equations which have $\displaystyle y $ on both sides, such as

$\displaystyle y=x^{\frac{y}{x}} $

If $\displaystyle y=x $, then this equation becomes

$\displaystyle y=x^{\frac{x}{x}} $

$\displaystyle y=x $

So from this I guess it can be concluded that one part of the graph of the equation $\displaystyle y=x^{\frac{y}{x}} $ is going to be the line $\displaystyle y=x $. However, there is another part to the graph since all of the inputs $\displaystyle x $ that are greater than 1 have two solutions for $\displaystyle y $ (except for $\displaystyle e $ apparently, according to the graph). Assuming $\displaystyle x=e $ is the only point that has one solution in the domain $\displaystyle x>1 $ of the graph $\displaystyle y=x^{\frac{y}{x}} $, then that is the point the graph intersects itself in a way.

My question basically is, how is it possible to solve for the value where such a graph (or graphs of this type in general) has only one solution (in other words how to find the point of its self-intersection), because in my example I assumed this point was $\displaystyle e $ because it really looked like it when I zoomed in. Is there a way to prove it without approximation?

Best,

Ivan