1. **State the problem:** Solve the equation $4^x = 2 \cdot \sqrt{2}$ for $x$. 2. **Rewrite the bases:** Note that $4 = 2^2$ and $\sqrt{2} = 2^{\frac{1}{2}}$. 3. **Express both sides with base 2:** $$4^x = (2^2)^x = 2^{2x}$$ $$2 \cdot \sqrt{2} = 2^1 \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2}} = 2^{\frac{3}{2}}$$ 4. **Set the exponents equal:** Since the bases are the same and nonzero, we have $$2x = \frac{3}{2}$$ 5. **Solve for $x$:** $$x = \frac{\frac{3}{2}}{2} = \frac{3}{2} \cdot \frac{1}{2} = \frac{3}{4}$$ **Final answer:** $$x = \frac{3}{4}$$