1. **State the problem:** Solve the equation $$4^{x-2} = \frac{1}{8}$$ for $x$. 2. **Recall the bases:** Note that $4$ and $8$ can be expressed as powers of $2$: $$4 = 2^2 \quad \text{and} \quad 8 = 2^3$$ 3. **Rewrite the equation using base 2:** $$4^{x-2} = (2^2)^{x-2} = 2^{2(x-2)}$$ $$\frac{1}{8} = 8^{-1} = (2^3)^{-1} = 2^{-3}$$ 4. **Set the exponents equal:** Since the bases are the same and nonzero, we can equate the exponents: $$2(x-2) = -3$$ 5. **Solve for $x$:** $$2x - 4 = -3$$ $$2x = 1$$ $$x = \frac{1}{2}$$ **Final answer:** $$x = \frac{1}{2}$$