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
$(x+7)+x+2 \sqrt{x^{2}+7 x}=49$
$\Rightarrow 2 x-42=-2 \sqrt{x^{2}+7 x}$
$\Rightarrow x-21=-\sqrt{x^{2}+7 x}$
Again, squaring both sides, we get
$x^{2}+441-42 x=x^{2}+7 x$
$\Rightarrow 49 x=441$
$\Rightarrow x=\frac{441}{49} $
$\Rightarrow x=9$