Question

The number of solutions of the equation log${}_{101}$log${}_7$($\sqrt{x+7}$ + $\sqrt{x}$) = 0 is

• A. 9
• B. 1
• C. 2
• D. 3

Answer 1

The correct option is A 9

Given,

${\log }_{101}\left({\log }_7(\sqrt{x+7}+\sqrt{x})\right)=0$

$\Rightarrow {\log }_7(\sqrt{x+7}+\sqrt{x})={101}^0$

$\Rightarrow \sqrt{x+7}+\sqrt{x}=7^{{101}^0}$

$\Rightarrow \sqrt{x+7}+\sqrt{x}=7$

$\Rightarrow \sqrt{x+7}=7-\sqrt{x}$

$\Rightarrow {\left(\sqrt{x+7}\right)}^2={(7-\sqrt{x})}^2$

$x+7=49-14\sqrt{x}+x$

$-42=-14\sqrt{x}$

$\Rightarrow 1764=196x$

$\therefore x\approx 9$