1. **State the problem:** Solve the equation $$249 \cdot 3^{x+2} = \frac{1}{9}$$ for $x$. 2. **Recall the properties of exponents:** - $3^{x+2} = 3^x \cdot 3^2$ - $\frac{1}{9} = 3^{-2}$ since $9 = 3^2$ 3. **Rewrite the equation using these properties:** $$249 \cdot 3^x \cdot 3^2 = 3^{-2}$$ 4. **Simplify the constants:** $$249 \cdot 9 \cdot 3^x = 3^{-2}$$ $$2241 \cdot 3^x = 3^{-2}$$ 5. **Isolate $3^x$:** $$3^x = \frac{3^{-2}}{2241}$$ 6. **Express the right side as a single power of 3 if possible:** Since $2241$ is not a power of 3, we keep it as is. 7. **Take the logarithm base 3 of both sides:** $$x = \log_3 \left( \frac{3^{-2}}{2241} \right) = \log_3(3^{-2}) - \log_3(2241) = -2 - \log_3(2241)$$ 8. **Final answer:** $$\boxed{x = -2 - \log_3(2241)}$$