Question

State True or False.

$5^{{\log }_{1/5}\left|1/2\right|}+{\log }_{\sqrt{2}}\frac{4}{\sqrt{7}+\sqrt{3}}+{\log }_{1/2}\frac{1}{10+2\sqrt{21}}$

Answer 1

$=5^{{\log }_{5}2}+{\log }_{\sqrt{2}}(\sqrt{7}-\sqrt{3})+{\log }_{\frac{1}{2}}\frac{10-2\sqrt{21}}{16}$
$=2+{\log }_{\sqrt{2}}(\sqrt{7}-\sqrt{3})+{\log }_{\frac{1}{2}}{\left(\frac{\sqrt{7}-\sqrt{3}}{4}\right)}^2$
$=2+{\log }_{\sqrt{2}}(\sqrt{7}-\sqrt{3})+2[{\log }_{\frac{1}{2}}(\sqrt{7}-\sqrt{3})-{\log }_{\frac{1}{2}}4]=6$
$=2+2{\log }_2(\sqrt{7}-\sqrt{3})-2{\log }_2(\sqrt{7}-\sqrt{3})+4$