finding value of x for an equation

Question

if we have an equation of form $y=x^{nx+1}$ and if we are given the values of $y$ and $n$ then how can one find $x$? I have reduced the equation to $\log(y)/\log(x)=nx+1$ but can't proceed further. Is there some kind of standard equation? Thanks

Answer 1

A quick search on Google reveals the page Link, where Doctor Vogler points out that the function $f(X) = x^x$ is not injective, since $$(\frac{1}{2})^{(1/2)} = (\frac{1}{4})^{(1/4)}.$$ However, he points out that it is possible to restrict the domain of the function so that it is injective. Nevertheless, I'm inclined to think that due to this observation, no notation may have been invented specifically for the inverse of this function, unlike other functions such as $f(x) = e^x$ which have inverses like $f^{-1}(x)=ln(x)$.

Answer 2

There is no solution in terms of the standard functions of calculus (powers, roots, logs, exponentials, trig functions, inverse trig functions). There are numerical methods for getting good approximate answers (the best ones rely on Calculus). It may be possible to express solutions in terms of the Lambert-W function, which you might want to look up.