Question

If ${\log }_{2}x+{\log }_{x}2=\frac{10}{3}={\log }_{2}y+{\log }_{y}2$ and $x=y$, then $x+y$ is equal to

• A. $2$
• B. $\frac{65}{8}$
• C. $\frac{37}{6}$
• D. none of these

Answer 1

The correct option is D none of these

${\log }_{2}x+{\log }_{x}2=\frac{10}{3}={\log }_{2}y+{\log }_{y}2$ ....(1)
and $x=y$
Equation (1) is valid when $x,y>0$ and $x,y\ne 1$

${\log }_{2}x+{\log }_{x}2=\frac{10}{3}$
$\Rightarrow {\log }_{2}x+\frac{1}{{\log }_{2}x}=\frac{10}{3}$
On substituting $t={\log }_{2}x$ in above equation, we get
$\Rightarrow 3t^2-10t+3=0$
$\Rightarrow t=3,\frac{1}{3}$
$\Rightarrow {\log }_{2}x=3,\frac{1}{3}$
$\Rightarrow x=8,\sqrt[3]{32}$
Since, $x=y$
Therefore, $x+y=16,2\sqrt[3]{32}$
Ans: D