7.088. $3 \cdot 4^{\log _{x} 2}-46 \cdot 2^{\log _{x} 2-1}=8$.
Solution.

Domain of definition: $0<x \neq 1$.

We have $3 \cdot 2^{2 \log _{x} 2}-23 \cdot 2^{\log _{x} 2}-8=0$. Solving the equation as a quadratic in terms of $2^{\log _{x} 2}$, we find $2^{\log _{x} 2}=-\frac{1}{3}, \varnothing$; or $2^{\log _{x} 2}=8$, from which $\log _{x} 2=3, x=\sqrt[3]{2}$.

Answer: $\sqrt[3]{2}$.