1. Given that $x$ satisfies $\log _{5 x} 2 x=\log _{625 x} 8 x$, then the value of $\log _{2} x$ is
Answer: $\frac{\ln 5}{2 \ln 2-3 \ln 5}$
Since $\log _{5 x} 2 x=\log _{625 x} 8 x$, so $\frac{\ln 2+\ln x}{\ln 5+\ln x}=\frac{3 \ln 2+\ln x}{4 \ln 5+\ln x}$
Therefore, $\ln x=\frac{\ln 2 \cdot \ln 5}{2 \ln 2-3 \ln 5}$
Thus, $\log _{2} x=\frac{\ln x}{\ln 2}=\frac{\ln 5}{2 \ln 2-3 \ln 5}$