How to Move the Intercepts of $x^y=y^x$ to (1, 1)

Question

In the equation, $x^y=y^x$, which I have modified to $(gx)^{y}=(gy)^{x}$, what would $g$ have to be to make the $f(x)=x$ function to intercept with the hyperbolic function at the point $(1,1)$?

$gx^y=gy^x$ where $g=1$">

Answer 1

The question is not quite clear to me. At point (x,y)=(1,1) or any other point where x=y, any real value of g would give equality to the expression $(gx)^y=(gy)^x$.

In the general case, where x is not necessarily equal y, this rounded value of g=2.71828 will cause equality of the equation and intercept with $y=x$ at point $(1,1)$, hence you get: $$(2.71828x)^{y}=(2.71828y)^{x}$$.

In fact, in this case the foollowing is true:

$$g=e$$

So we have:

$$(e.x)^{y}=(e.y)^{x}$$.

Where e is the e-The Famous Math Constant

Since you are using Desmos, maybe you don't want an analytic/algebraic detailed solution and the graph would suffice.

See: Desmos Plot