Good evening,

Given $\alpha_1,\dots,\alpha_n$ non-zero algebraic numbers. I want to find an effective bound $N$ such that if for all $(\beta_1,\dots,\beta_n)\in\mathbb{Z}^n\setminus \{0\}$ with $|\beta_i| \leq N,\; i=1,\dots, n,$ we have $\beta_1 \ln \alpha_1 +\dots + \beta_n \ln \alpha_n \neq 0$, then $\ln \alpha_1,\dots,\ln \alpha_n$ are linearly independent over $\mathbb{Q}$.

I thought it would be possible to use Baker’s Theorem, but I don't see how. It's probably an already known result, so if you find a reference, it's also good to me. Thank you in advance.