$\lim_{n\to\infty} \frac{(\log n)^n}{n^{\log n}}$ $$\lim_{n\to\infty} \frac{(\log n)^n}{n^{\log n}}$$

This limit is equal to $+\infty$ but I don't understand why. I know that every exponential function with a base greater than 1 grows to infinity faster than any polynomial. But is this true only for a fixed base exponential that doesn't depend on the variable?

Or is it that, approaching positive infinity, the function at the numerator behaves like $a^n, a &gt; 1$ in terms of comparison with a polynomial?

Sorry if this is a silly question.
By taking the logarithm
$$\log\left(\frac{(\log n)^n}{n^{\log n}}\right)=n\log\log n-\log^2 n$$

and then rewriting
$$n\log\log n-\log^2 n= n\left(\log\log n-\frac{\log^2n }{n}\right)$$
In the parenthesis, the second term goes to $0$. What happens to the first?
$6^{1000000} &gt;&gt; 1000000^6 \ $ because that's a lot of $6$'s