Is there a way to express the geometric mean in terms of the $\mathrm{L}^p$ norms?

Question

Sorry, I can't find a good tag for this.

So the "scaled" $\mathrm{L}^p$ norm is:

$$ \| \mathbf{x} \| = \left( \frac{1}{n} \sum\limits_{j=1}^{n} |x_j|^p \right)^{1/p} $$

and the geometric mean is:

$$ \overline{\mathbf{x}} = \exp \left( \frac{1}{n} \sum\limits_{j=1}^{n} \log(x_j) \right) $$

Is there an elegant way to express the latter in terms of the former?