1. **State the problem:** Solve the equation $$3^x = \frac{1}{27}$$ for $x$. 2. **Recall the properties of exponents:** We know that $27 = 3^3$, so $$\frac{1}{27} = 3^{-3}$$ because $\frac{1}{a^n} = a^{-n}$. 3. **Rewrite the equation using the same base:** $$3^x = 3^{-3}$$ 4. **Since the bases are equal and nonzero, set the exponents equal:** $$x = -3$$ 5. **Final answer:** $$x = -3$$