Question

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

• A. 3
• B. 7
• C. 9
• D. 49

Answer 1

Answer: (C)

Given, ${\log }_{101}{\log }_7(\sqrt{x+7}+\sqrt{x})=0$
$\therefore {\log }_7(\sqrt{x+7}+\sqrt{x})=(101{)}^0$
$\Rightarrow {\log }_7(\sqrt{x+7}+\sqrt{x})=1$
$\Rightarrow (\sqrt{x+7}+\sqrt{x})=7^1$
$\Rightarrow \sqrt{x+7}+\sqrt{x}=7$
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
$x^2+441-42x=x^2+7x$
$\Rightarrow 49x=441$
$\Rightarrow x=\frac{441}{49}$
$\Rightarrow x=9$