$\text{ The value of }{5}^{{\log }_{1/5}(1/2)}+{\log }_{\sqrt{2}}\frac{4}{\sqrt{7}+\sqrt{3}}+{\log }_{1/2}\frac{1}{10+2\sqrt{21}}\text{ is }$

$\begin{array}{l}{5}^{{\log }_{1/5}(1/2)}+{\log }_{\sqrt{2}}\frac{4}{\sqrt{7}+\sqrt{3}}+{\log }_{1/2}\frac{1}{10+2\sqrt{21}}\\ =2+{\log }_2\left\{{\left(\frac{4}{\sqrt{7}+\sqrt{3}}\right)}^2(\sqrt{7}+\sqrt{3}{)}^2\right\}\\ =2+{\log }_{2}16=6\end{array}$