I would like to know the purpose of having quantitative bounds in transcendental number theory. In particular, with the help of examples, I would like to know how to interpret these statements as transcendence results and in what other ways, are these quantitative results helpful.
For instance, the quantitative version of Baker's theorem tells us that If $\lambda_1,…,\lambda_n$ are non zero logarithms of algebraic numbers that are linearly independent over the rational numbers, then for any algebraic numbers $\beta_0, ..., \beta_n$, not all zero, we have $ \\ |\beta _{0}+\beta _{1}\lambda _{1}+\cdots +\beta _{n}\lambda _{n}|>H^{{-C}} \\$ where H is the maximum of the heights of the $\beta$'s and C is an effectively computable number depending on n, the $\lambda$'s, and the maximum d of the degrees of the $\beta$'s. In this case, the only conclusion I can make is that, if the logarithms of algebraic numbers are linearly independent over $\mathbb{Q}$, they are linearly independent over $\overline{\mathbb{Q}}$. Is there any other conclusion that we can make from the above quantitative version?