1. **State the problem:** Solve for $x$ in the equation $4^x = 256$. 2. **Recall the formula and rules:** We know that $a^m = a^n$ implies $m = n$ if the bases $a$ are the same. 3. **Express both sides with the same base:** Since $4 = 2^2$ and $256 = 2^8$, rewrite the equation as: $$4^x = 256 \implies (2^2)^x = 2^8$$ 4. **Simplify the left side using power of a power rule:** $$(2^2)^x = 2^{2x}$$ So the equation becomes: $$2^{2x} = 2^8$$ 5. **Set the exponents equal:** $$2x = 8$$ 6. **Solve for $x$:** $$x = \frac{8}{2} = 4$$ **Final answer:** $$x = 4$$