Question

If '$S$' is sum of all real values of '$x$' satisfying the equation $lo{g}_{1/2}\left(lo{g}_2\right(x+\sqrt{5})+lo{g}_2(x-\sqrt{5})+lo{g}_{1/2}(x-2\left)\right)-1$, then '$S$' is equal to

• A. a rational number
• B. a whole number
• C. a prime number
• D. neither prime nor composite

Answer 1

The correct option is A a rational number

Given,

$lo{g}_{\frac{1}{2}}\left(lo{g}_2\right(x+\sqrt{5})+lo{g}_2(x-\sqrt{5})+lo{g}_{\frac{1}{2}}(x-2\left)\right)-1=0$

$log\left(lo{g}_2\right(x^2-5)-lo{g}_2(x-2\left)\right)=log\frac{1}{2}$

$lo{g}_2\left(\frac{x^2-5}{x-2}\right)=\frac{1}{2}$

$log(\frac{x^2-5}{x-2}=log{2}^{\frac{1}{2}}$

$⟹\frac{x^2-5}{x-2}=\sqrt{2}$

$⟹x^2-5=\sqrt{2}x-2\sqrt{2}$

$⟹x^2-\sqrt{2}x+2\sqrt{2}-5=0$

$⟹x=\frac{\sqrt{2}\pm \sqrt{2-4(2\sqrt{2}-5)}}{2}$

Hence the sum of values of $x$ is in the form of $\frac{p}{q}$ So It must be a rational number.