Write a value

let $\log x^{n}=t$

Differentiating on both sides we get,

$\frac{1}{x^{n}} n x^{n-1} d x=d t$

$\frac{n}{x} d x=d t$

$\frac{1}{x} d x=\frac{1}{n} d t$

Substituting above equations in $\int \frac{\log x^{n}}{x} d x$ we get,

$\int \frac{1}{n} t d t$

$=\frac{1}{n} \frac{t^{2}}{2}+c$

$=\frac{\left(\log x^{n}\right)^{2}}{2 n}+c$