Linear Forms in the Logarithms of Algebraic Numbers with Small Coefficients I

Ifav α₁, α₂, β₁ are rational numbers satisfying (i) α₁ > 0, α₂ > 0 are multiplicatively independent (ii) the size of α₁, α₂, β₁, respectively, do not exceed S₁, S₁ and (log S₁)¹⁰⁰ (100 is quite unimportant), then | β₁ log α₁ - log α₂| > C(∈) exp ( - (log S₁)^2+z) (1) where ∈ > 0 is an arbitrary fixed constant and C(∈) is an effectively computable positive constant depending only on ∈.