L’Hôpital’s rule practice question 12

What does each fraction do as $x \to 1$? The rule works on a single quotient, so how could you make one from this difference?

The form is $\infty - \infty$. Combine over a common denominator to get $\dfrac{x - 1 - \log x}{(x - 1)\log x}$, which is $\tfrac{0}{0}$.

Two applications are needed; simplify between them by multiplying through by $x$.