Question

Simplify: $5^{{\log }_{\frac{1}{5}}\frac{1}{2}}+{\log }_{\sqrt{2}}\left(\frac{4}{\sqrt{7}+\sqrt{3}}\right)+{\log }_{\frac{1}{2}}\left(\frac{1}{10+2\sqrt{21}}\right)$

Answer 1

let $y=5^{{\log }_{\frac{1}{5}}\left(\frac{1}{2}\right)}+{\log }_{\sqrt{2}}\left(\frac{4}{\sqrt{7}+\sqrt{3}}\right)+{\log }_{\frac{1}{2}}\left(\frac{1}{10+2\sqrt{21}}\right)$
$\Rightarrow y=5^{{\log }_{5}2}[\because {\log }_{\frac{1}{a}}\frac{1}{b}={\log }_{a}b]+2{\log }_2\left(\frac{4}{\sqrt{7}+\sqrt{3}}\right)[\because {\log }_{a^m}b=\frac{1}{m}{\log }_{a}b]-{\log }_2\left(\frac{1}{{(\sqrt{7}+\sqrt{3})}^2}\right)$
$\Rightarrow y=2+2{\log }_2{2}^2-2{\log }_2(\sqrt{7}+\sqrt{3})+2{\log }_2(\sqrt{7}+\sqrt{3})[\because \log a^m=m\log a]$
$\Rightarrow y=2+4{\log }_{2}2=6,[\because {\log }_{a}a=1]$
Ans: 6