Solve: \(\left(\frac{3}{5}\right)^x \left(\frac{5}{3}\right)^{2x} = \frac{125}{27}\)
Solution
\[ \left(\frac{3}{5}\right)^x \left(\frac{5}{3}\right)^{2x} = \frac{125}{27} \]
\[ \Rightarrow \frac{3^x}{5^x} \cdot \frac{5^{2x}}{3^{2x}} = \frac{125}{27} \]
\[ \Rightarrow \frac{3^x \cdot 5^{2x}}{5^x \cdot 3^{2x}} = \frac{125}{27} \]
\[ \Rightarrow \frac{5^x}{3^x} = \frac{125}{27} \]
\[ \Rightarrow \left(\frac{5}{3}\right)^x = \left(\frac{5}{3}\right)^3 \]
\[ \Rightarrow x = 3 \]
Final Answer:
\[ \boxed{x = 3} \]