Question

If $x=2^{\sqrt{5}}{.5}^{\sqrt{2}}$ and ${\log }_{10}x=(\sqrt{P}-\sqrt{Q}){\log }_{10}2+\sqrt{Q},$ then $P+Q=$

• A. $11$
• B. $9$
• C. $7$
• D. $13$

Answer 1

The correct option is C $7$

$x=2^{\sqrt{5}}{.5}^{\sqrt{2}}$

${\log }_{10}x={\log }_{10}\left(2^{\sqrt{5}}{.5}^{\sqrt{2}}\right)$

$={\log }_{10}{2}^{\sqrt{5}}+{\log }_{10}{5}^{\sqrt{2}}$

$=\sqrt{5}{\log }_{10}2+\sqrt{2}{\log }_{10}5$

$=\sqrt{5}{\log }_{10}2+\sqrt{2}\left({\log }_{10}\left(\frac{10}{2}\right)\right)$

$=\sqrt{5}{\log }_{10}2+\sqrt{2}({\log }_{10}10-{\log }_{10}2)$

$=\sqrt{5}{\log }_{10}2+\sqrt{2}(1-{\log }_{10}2)$

$=(\sqrt{5}-\sqrt{2}){\log }_{10}2+\sqrt{2}$

$\therefore P=5$ and $Q=2$

$\therefore P+Q=7$