Plotting nth root of x against nx on a graph.

Question

I've spent the last two days trying to figure this out.

What I'm trying to do is rearrange this:

$$ x^n = \frac xn $$

to make n the subject, to allow me to plot on a graph, with n being the y-value and x being itself.

Answer 1

Let's assume $x,n$ are each positive. Multiply both sides by $n$ and divide by $x$ to get $$nx^{n-1}=1$$ Now take logs to get $\log n + (n-1) \log x=0$, which rearranges to $$\log x=\frac{-1}{n-1}\log n$$ Now exponentiate both sides to get the answer $$x=n^{-1/(n-1)}=\frac{1}{\sqrt[n-1]{n}}$$