TITLE: How to Move the Intercepts of $x^y=y^x$ to (1, 1)
QUESTION [2 upvotes]: 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 $g=1$">

REPLY [1 votes]: 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
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