Why does the graph $y=(\ln y/\ln x)x (-\ln y/\ln x (x^2/y))+x$ not include all positive $y$ and $x$ for which $y = x$? When I plotted this graph on desmos , the plot did not include the line $y = x$, even though the equation is solved for all $y = x$. This equation came up in the context of me trying to prove that the equations of the solutions to $x^y = y^x$ intersect at the point $(e,e)$
Why does graph $y=(\ln y/\ln x) x-\ln y/\ln x (x^2/y)+x$ not include $x = y$
algebra-precalculusfunctions
Best Answer
If we substitute $y=x$ in the equation $$y=-\frac{x^2 \ln y}{y \ln x}+\frac{\ln y}{\ln x}x+x$$ we get $x=-x+x+x\to 0=0$ undefined.
Thus the equation above is not equivalent to $$x^y=y^x\tag{1}$$
I strongly recommend to simply prove by direct inspection that $x=y$ is a set of solution of $(1)$, and use this answer to prove that $(e,e)$ is a self intersection.
Hope this helps