Finite dimensional $l^p$ Norm is logarithmically convex.

I am working trough this book, where in lemma 1.6 (proof left to the reader), the asked to show that the map $p \mapsto \log \|X\|_{p,J}^p$ is convex. Here $\|X\|_{p,J}$ is the $p$-variation of a continuous path for $p \geq 1$.

To show this, it should be sufficient to show that the for any $x \in \mathbb{R}^n$ the map $p \mapsto \log\|x\|_p^p$ is convex, where $\|x\|_p = \left(\sum_{i=1}^n |x_i|^p\right)^{\frac1p}$.

However, I cannot seem to figure out how to show this, simply differentiating twice did not work (the expression got two complicated, as far as I can see it).

Can someone give me a hint on how to start proving the above statement? Any link or reference would also be appreciated.

Thanks in advance!