ABC26GN2082 · Logarithms

$$\begin{aligned} & \log _{7} \log _{5}(\sqrt{x+5}+\sqrt{x})=0 \Rightarrow \log _{5}(\sqrt{x+5}+\sqrt{x})=7^{0}=1 \\ & \Rightarrow \sqrt{x+5}+\sqrt{x}=5^{1}=5 \Rightarrow(\sqrt{x+5}+\sqrt{x})^{2}=25 \\ & \Rightarrow(x+5)+x+2 \sqrt{x+5} \sqrt{x}=25 \Rightarrow 2 x+2 \sqrt{x} \sqrt{x+5}=20 \\ & \Rightarrow \sqrt{x} \sqrt{x+5}=10-x \Rightarrow x(x+5)=(10-x)^{2} \\ & \Rightarrow x^{2}+5 x=100+x^{2}-20 x \Rightarrow 25 x=100 \Rightarrow x=4 \end{aligned}$$