Question

Show that
${\log }_y\sqrt{x}.{\log }_z{y}^3.{\log }_x\sqrt[3]{3z^2}=1$

Answer 1

LHS$={\log }_y\sqrt{x}.{\log }_z{y}^3.{\log }_x{z}^{2/3}$

$=\frac{1}{2}\left({\log }_{y}x\right).3{\log }_{z}y.\frac{2}{3}{\log }_{x}z$

$=\frac{\log x}{\log y}.\frac{\log y}{\log z}.\frac{\log z}{\log x}$

$=1$

=RHS

Hence proved.