2. Solve the equation $\left(\frac{3 x}{2}\right)^{\log _{3}(8 x)}=\frac{x^{7}}{8}$.
Answer: $x=\frac{729}{8}, x=2$.

Solution. By taking the logarithm to base 3, we obtain an equation equivalent to the original:

$$
\log _{3}(8 x) \cdot \log _{3}\left(\frac{3 x}{2}\right)=\log _{3}\left(\frac{x^{7}}{8}\right) \Leftrightarrow\left(\log _{3} x+3 \log _{3} 2\right)\left(1+\log _{3} x-\log _{3} 2\right)=7 \log _{3} x-3 \log _{3} 2
$$

Let $\log _{3} x=y, \log _{3} 2=a$. After expanding the brackets and combining like terms, the equation becomes $y^{2}+(2 a-6) y-3 a^{2}+6 a=0$. We solve the quadratic equation for $y$:

$$
\frac{D}{4}=(a-3)^{2}+\left(3 a^{2}-6 a\right)=4 a^{2}-12 a+9=(2 a-3)^{2}, y=3-a \pm(2 a-3) \Leftrightarrow\left[\begin{array}{l}
y=a \\
y=6-3 a
\end{array}\right.
$$

We find $x$: if $y=a$, then $\log _{3} x=\log _{3} 2$ and $x=2$; if $y=6-3 a$, then $\log _{3} x=6-3 \log _{3} 2$ and $x=\frac{729}{8}$.