7.71 $\sqrt{\log _{x} \sqrt{5 x}}=-\log _{x} 5$.
7.71 Since the left side of the equation is non-negative (arithmetic root), the inequality $-\log _{x} 5>0$ must be satisfied, i.e., $\log _{x} 5<0$, from which $0<x<1$. Squaring both sides of the equation:

$\log _{x} \sqrt{5 x}=\log _{x}^{2} 5$, or $\frac{1}{2} \log _{x} 5+\frac{1}{2}=\log _{x}^{2} 5$.

Let $\log _{x} 5=y<0$, we get the equation $2 y^{2}-y-1=0$, which has roots $y_{1}=-0.5, y_{2}=1$ (does not fit since the condition $y<0$ is not satisfied). Therefore, $\log _{x} 5=-0.5$, from which $x^{-0.5}=5$, i.e., $x=5^{-2}=0.04$.

Answer: $x=0.04$.