7.079. $2 \log _{x} 27-3 \log _{27} x=1$.
Solution.

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

Let's switch to base 27. We have

$$
\frac{2}{\log _{27} x}-3 \log _{27} x-1=0 \Rightarrow 3 \log _{27}^{2} x+\log _{27} x-2=0
$$

Solving this equation as a quadratic equation in terms of $\log _{27} x$, we get $\left(\log _{27} x\right)_{1}=-1$, from which $x_{1}=\frac{1}{27}$, or $\left(\log _{27} x\right)_{2}=\frac{2}{3}$, from which $x_{2}=27^{\frac{2}{3}}=9$. Answer: $\frac{1}{27} ; 9$.