Solving equations of type $x^{1/n}=\log_{n} x$ First, I'm a new person on this site, so please correct me if I'm asking the question in a wrong way.
I thought I'm not a big fan of maths, but recently I've stumbled upon one interesting fact, which I'm trying to find an explanation for. I've noticed that graphs of functions $y = x^{1/n}$ and $y = \log_{n} x$ , where $n$ is given and equal for both functions, always have $2$ intersection points. This means, equation  $x^{1/n}= \log_{n} x$ must have $2$ solutions, at least it's what I see from the graphs.
I've tried to solve this equation analytically for some given $n$, like $4$, but my skills are very rusty, and I cannot come up with anything. So I'm here for help, and my question(-s) are:


*

*are these $2$ functions always have $2$ intersection points?

*if yes, why, if not, when not?

*how to solve equations like $x^{1/n}= \log_{n} x$ analytically?


$$\begin{align}\sqrt[n]x&=\log_nx\\\\x&=t^n\end{align}\ \Bigg\}\iff t=\frac n{\ln n}\cdot\ln t\quad;\quad t=e^u\iff e^u=\frac n{\ln n}\cdot u\iff$$
$$(-u)\cdot e^{-u}=-\frac{\ln n}n\iff u=-W\bigg(-\frac{\ln n}n\bigg)\iff x=t^n=(e^u)^n=e^{nu}$$
$$x=\exp\bigg(-n\cdot W\bigg(-\frac{\ln n}n\bigg)\bigg)$$ where W is the Lambert W function.
