Solution to an open problem on a logarithmic integral and derived results

This article solves an open problem that was previously stated by providing an exact evaluation of a logarithmic integral. Furthermore, the result is generalized by introducing a new adjustable parameter. Based on this extension, other integral formulas are derived. Applications are presented for integral inequalities of Hölder and Hardy-Hilbert types, in which the introduced parameter plays a pivotal role in establishing the bounds under suitable integrability conditions.