Question: I want to solve that equation for $x$, given a natural number $n$.

Can someone help?

Thanks!

Answer: 
Find the natural domain of $f(x)=\tan(nx)+\log n-\frac{1}{x}$. Note it is differentiable on its domain. Compute the derivative. Study its sign. Deduce the variations on each interval of the domain. Apply the intermediate value theorem and observe strict monotonicity to find a unique zero on each of these intervals. There is no closed form.

**Extra help:** [click here](https://www.desmos.com/calculator/mzwax9gtyy) to see the graph when $n=2$.
I assume your question is $${1\over x}-\tan(nx)=\log n$$ In general, there is no way to find a solution in terms of the standard functions. You can find a numerical approximation as accurate as you like by any of a number of methods, of which the most familiar is probably Newton's Method, from intro Calculus.