Question

If $x=\frac{\sqrt{15}+\sqrt{7}}{2}$ and $y=\frac{\sqrt{15}-\sqrt{7}}{2}$, then the value of ${\log }_3(x^2+y^2-xy)$ is

Answer 1

${\log }_3(x^2+y^2-xy)$
$\Rightarrow {\log }_3\left(\right(x-y{)}^2+xy)$
$x=\frac{\sqrt{15}+\sqrt{7}}{2}$
$y=\frac{\sqrt{15}-\sqrt{7}}{2}$
$(x-y{)}^2=7$
$xy=2$
$\Rightarrow {\log }_3\left(\right(x-y{)}^2+xy)={\log }_3(7+2)={\log }_{3}9=2$