If _7 _5 ( x + 5 + x ) = 0, what is the value of x ?

Answer: Option C

$$\begin{array}{rl}& {\log }_7{\log }_5\left(\sqrt{x+5}+\sqrt{x}\right)=0\\ & \Rightarrow {\log }_5\left(\sqrt{x+5}+\sqrt{x}\right)=7^0=1\\ & \Rightarrow \sqrt{x+5}+\sqrt{x}=5^1=5\\ & \Rightarrow {\left(\sqrt{x+5}+\sqrt{x}\right)}^2=25\\ & \Rightarrow \left(x+5\right)+x+2\sqrt{x+5}\sqrt{x}=25\\ & \Rightarrow 2x+2\sqrt{x}\sqrt{x+5}=20\\ & \Rightarrow \sqrt{x}\sqrt{x+5}=10-x\\ & \Rightarrow x\left(x+5\right)={\left(10-x\right)}^2\\ & \Rightarrow x^2+5x=100+x^2-20x\\ & \Rightarrow 25x=100\\ & \Rightarrow x=4\end{array}$$