Let $ \alpha_1, \ldots, \alpha_r $ be elements of a field $K \supseteq \mathbb Q$, which have the property that for $n_i \in \mathbb Z$, $i=1,\ldots,r$, it follows from $\alpha_1^{n_1}\cdots \alpha_r^{n_r}=1$ that $n_i=0 \quad \forall \ i=1,\ldots,r$.
How does one prove that from $\beta_1\log_p(\alpha_1) + \cdots + \beta_r\log_p(\alpha_r)=0\ $ with $ \beta_i \in \mathbb Q $ it follows: $ \beta_i = 0\ \forall\ i=1,\ldots,r $ ?
($\log_p$ denotes the $p$-adic logarithm)