how would you find the inverse of this function

[Notta]

if you consider the equation x^y = y^x and graph it, you have the trivial solution of y=x , but you also have another part (or rather 2 parts) to the graph. you'll see that (2,4) and (4,2) are solutions, yet they don't fall on the line y=x. they do satisfy a different equation, though. specifically, you can get (4,2) by plugging in 4 to the function f(x) = -x/ln(x) * W(-ln(x)/x) where W(x) is the Lambert W function. this function in particular is the exact function for every non-trivial solution where x>e. for x<e, the function describing it is the inverse of f(x). please help me find the inverse of f(x)

desmos graph if you want to see hands on