I denote $p_k$ the $k^{th}$ prime number ($p_1=2$, etc...)

Clearly, for any $n\in \mathbb{N}^*$, $(\log p_k)_{1\leq k\leq n}$ is linearly independent over $\mathbb{Q}$.

My question concerns a possible generalization: is it true that for any $q,n\in \mathbb{N}^*$, $((\log{p_k})^q)_{1\leq k\leq n}$ is linearly independent over $\mathbb{Q}$ ?

I am of course interested in weaker statements : $q=2$, "for any $n$ there exists $q\geq 2$ such that..." or "there exists $q\geq 2$ such that for any $n$...".

I am not an expert in the field, sorry if this is actually trivial.