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?


The second solution is explicit in the sense that it can be expressed as
x = [W(z) / z]^n
W noting Lambert function and z = - Ln(n) / n
I use on purpose the notation Ln for natural logarithm.  
What is extremely amazing is that, at least for 4 < n < 13, the value of the second solution varies linearly with n [x = 10.6844 + 1.2821 n ; correlation coefficient greater than 0.99997 !] .  
Sorry for not being able to justify that.  
Thanks for the nice problem and Merry Xmas to everyone.
