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

The purpose of this note is to prove the following:

THEOREM 1. If α₁, α₂, α₃, β₁, β₂ are rational numbers satisfying (i) α₁ > 0, α₂ > 0, α₃ > 0 are multiplicatively independent (ii) the size of α_i, ≤ S₁, i = 1, 2, 3, and that of β_i, ≤ (log S₁)¹⁰⁰ = S, i = 1, 2, (100 is quite unimportant), then

| β₁ log α₁ + β₂ log α₂ - log α₃ | > C(∈) exp ( - (log ₁)^8+∈)

where ∈ > 0 is an arbitrary fixed constant and C(∈) is an effectively computable positive constant depending only on ∈.