Baker's theorem for integer combinations of logarithms of integers? Baker's theorem in transcendental number theory states that
$$
\left|\beta_0 +  \sum_{i=1}^n \beta_i \log \alpha_i\right| > H^{-C}
$$
where

*

*$\beta_0, \ldots, \beta_n$ are algebraic numbers, not all zero,

*$\alpha_1, \ldots, \alpha_n$ are multiplicatively independent algebraic numbers,

*$H$ is the maximum of the heights of $\beta_i$, and

*$C$ is an effectively computable constant depending on $n$, $\alpha_i$, and the maximum of the degrees of $\beta_i$.

Is there a simple proof for a special case of Baker's theorem where $\beta_0 = 0$, the $\beta_i$ are rational integers, and the $\alpha_i$ are positive rational integers?
 A: The main difficulty in proving Baker's theorem is in estimating $C$. If you don't care about those estimates, then the proof is not difficult. For example, Chapter 7 of Waldschmidt's Diophantine approximation on linear groups gives a simple proof for the homogeneous case $\beta_0=0$. Some technicalities (like using Fel'dman integer-valued polynomials) are not necessary if you don't care about strong estimates.
The homogeneous case is slightly simpler because one can work with just $G_m^n$ or (dually) with $G_a^n$. However, the basic structure of the proof is the same:

*

*Using group structure, consider several points (working with $G_m^n$, the points are $(\alpha_1^s,\dots,\alpha_n^s)$).

*construct an auxiliary function/polynomial either using Dirichlet principle (in the form of Siegel's lemma) or using interpolating determinant

*Show that this polynomial with some derivatives is small (using either Schwarz's lemma or Hermite's formula) at those points.

*Show that it can't vanish at all those points.

*Show that if the polynomial or its derivative is small at one of those points, the value must be zero.
Assumptions, that $\beta_i$ and $\alpha_i$ are integers simplify the last step - instead of Liouville's inequalities, you are just using directly that if $x$ is an integer with $|x|<1$ then $x=0$. However, the other steps are the same.

Baker's theorem can also be proved using the Schnieder-Lang criterion, but I don’t know if that method will provide estimates sufficient for your needs.
