Upper-bounding $\exp \log^{d} \frac{\log n}{n}$

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How would you upper-bound this expression? $$f(n, d) = \exp \log^{d} \frac{\log n}{n}$$

If $d = 1 $ this woulld simplify to $\frac{\log n}{n}$.
Any suggestions on how to upperbound it?

Notation clarification: $\log^x y = (\log y)^x$.