Question

Number of solutions satisfying, $\sqrt{5-{log}_{2}x}=3-{log}_{2}x$ are :

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The correct option is A 1

$\sqrt{5-{\log }_{2}x}=3-{\log }_{2}x$ ---- ( 1 )

Let ${\log }_{2}x=y$ ----- ( 2 )

$\Rightarrow$ $\sqrt{5-y}=3-y$

Squaring both sides,

$\Rightarrow$ $5-y=9-6y+y^2$

$\Rightarrow$ $y^2-5y+4=0$

$\Rightarrow$ $y^2-4y-y+4=0$

$\Rightarrow$ $y(y-4)-1(y-4)=0$

$\Rightarrow$ $(y-4)(y-1)=0$

$\Rightarrow$ $y=4$ and $y=1$

Substituting $y=1$ in ( 1 ) we get,

$\Rightarrow$ ${\log }_{2}x=1$

We know, ${\log }_{b}a=x\Rightarrow a=b^x$

$\therefore$ $x=2^1=2$

Substituting $y=4$ in ( 1 ) we get,

$\Rightarrow$ ${\log }_{2}x=4$

We know, ${\log }_{b}a=x\Rightarrow a=b^x$

$\therefore$ $x=2^4=16$

Substituting ${\log }_{2}x=1$ in ( 1 ) we get,

$\sqrt{5-1}=3-1$

$\Rightarrow$ $\sqrt{4}=2$

$\therefore$ $2=2$

Substituting ${\log }_{2}x=4$ in ( 1 ) we get,

$\sqrt{5-4}=3-4$

$\Rightarrow$ $\sqrt{1}=-1$

$\therefore$ $1=-1$

Hence, we can see only ${\log }_{2}x=1$ satisfying.

$\therefore$ $\sqrt{5-{\log }_{2}x}=3-{\log }_{2}x$ has only $1$ solution.