1. **State the problem:** Solve the exponential equation $$4^x = 2^6$$ for $$x$$. 2. **Rewrite bases as powers of the same base:** Note that $$4 = 2^2$$, so rewrite the equation as $$\left(2^2\right)^x = 2^6$$. 3. **Apply the power of a power rule:** $$\left(2^2\right)^x = 2^{2x}$$, so the equation becomes $$2^{2x} = 2^6$$. 4. **Set exponents equal:** Since the bases are the same and nonzero, $$2x = 6$$. 5. **Solve for $$x$$:** $$x = \frac{6}{2}$$. 6. **Simplify:** $$x = 3$$. **Final answer:** $$x = 3$$.