On The Irrationality Measure Of Logarithms Of Rational Numbers

Let α∈R/Q, If for any e>0, there exists q≥q0(ε), such that|α-p/q|≥q-μ-ε, for all integers p and q with q≥q0(e), then the real number μ>0is said to be an irrationality measure of α. The minimum of these numbers μ is denoted by μ(α).Let α0,α1…,αn be real numbers linearly independent over Q, If for any ε>0, there existsq≥q0(ε), such that|pα0+q1α1+…+qnαn|≥H-v-ε, for all integers p and qi with H=max(|q1|,|q2|,…,|qn|)≥H0(ε), then the real number v≥0is said to be a linear independence measure of α0,α1,…,αn.The minimum of these numbers v is denoted by v(α0,α1,…,αn). We remark that, with this notation, μ(α)=v(1,α)+1.Let a∈Z, In this paper, we mainly discussed the irrationality measure of the Logarithms of rational numbers which are log a+b/a-b. We have the irrationality measures for b=1, a≥3and for b=2, a=2m+1where m≥5respectively. And we have also the linear independence measures of the1, log(1-1/a), log(1+1/a) for a>2.