Question

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

• A. $3$
• B. $7$
• C. $9$
• D. $49$

Answer 1

The correct option is C $9$

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

$\therefore {\log }_7(\sqrt{x+7}+\sqrt{x})={\left(101\right)}^0$
$\Rightarrow {\log }_7(\sqrt{x+7}+\sqrt{x})=1$
$\Rightarrow (\sqrt{x+7}+\sqrt{x})=7^1$
$\Rightarrow \sqrt{x+7}+\sqrt{x}=7$

On squaring both sides, we get
$(x+7)+x+2\sqrt{x^2+7x}=49$
$\Rightarrow 2x-42=-2\sqrt{x^2+7x}$
$\Rightarrow x-21=-\sqrt{x^2+7x}$

Again squaring both sides, we get
$x^2+441-42x=x^2+7x$
$\Rightarrow 49x=441$
$\Rightarrow x=\frac{441}{49}$

$\Rightarrow x=9$