Approximating $\ln 2$ by numbers in the field $\mathbb{Q}(\sqrt{2}\,)$

Using a new integral construction combining the idea of symmetry suggested by the second author in 2007 and the integral introduced by Marcovecchio in 2009, we obtain a new bound for the approximation of $\ln{2}$ by numbers in the field $\mathbb{Q}(\sqrt{2}\,)$.