On the approximation of logarithms of algebraic numbers

Abstract

A new identity is given by means of which infinitely many algebraic functions approximating the logarithmic function In x are obtained. On substituting numerical algebraic values for the variable, a lower bound for the distance of its logarithm from variable algebraic numbers is found. As a further application, it is proved that the fractional part of the number e ^a is greater than a -40 a for every sufficiently large positive integer a .