### 7.340

$$
\left\{\begin{array}{l}
x^{\log _{3} y}+2 y^{\log _{3} x}=27 \\
\log _{3} y-\log _{3} x=1
\end{array}\right.
$$
## Solution.

$$
\left\{\begin{array} { l } 
{ x ^ { \log _ { 3 } y } + 2 x ^ { \log _ { 3 } y } = 27 , } \\
{ \log _ { 3 } y - \log _ { 3 } x = 1 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
x^{\log _{3} y}=9 \\
\log _{3} y=\log _{3} x+1
\end{array}\right.\right.
$$

Logarithmize the first equation:

$$
\begin{aligned}
& \left\{\begin{array} { l } 
{ \log _ { 3 } y \cdot \log _ { 3 } x = \log _ { 3 } 9 , } \\
{ \log _ { 3 } y = \log _ { 3 } x + 1 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
\log _{3}^{2} x+\log _{3} x-2=0, \\
\log _{3} y=\log _{3} x+1
\end{array} \Leftrightarrow\right.\right. \\
& \Leftrightarrow\left[\begin{array} { l } 
{ \log _ { 3 } x = - 2 , \log _ { 3 } y = - 1 , } \\
{ \log _ { 3 } x = 1 , \log _ { 3 } y = 2 }
\end{array} \Leftrightarrow \left[\begin{array}{l}
x=\frac{1}{9}, y=\frac{1}{3} \\
x=3, y=9
\end{array}\right.\right.
\end{aligned}
$$

Answer: $\left(\frac{1}{9} ; \frac{1}{3}\right),(3 ; 9)$.

## IDENTICAL TRANSFORMATIONS OF TRIGONOMETRIC EXPRESSIONS

Everywhere in this chapter, the left-hand side of the given identity in the problem will be denoted by $L$, and the right-hand side by $R$.

## Prove the identities /3.396-3.409/: