An integral with $\frac{1}{\log x}$

Suppose $n$ is a positive integer (though the result below does not really need that).

Show that

$$\int_{0}^{1} \frac{x^n - 1}{\log x} \text{d}x = \log(n+1)$$

Note that the $\log x$ is the $\log$ to base $e$.