1. **State the problem:** Solve the equation $$64^{2x-1} = 128^x \cdot 2^{x-1}$$ for $x$. 2. **Rewrite bases as powers of 2:** - $64 = 2^6$ - $128 = 2^7$ So the equation becomes: $$\left(2^6\right)^{2x-1} = \left(2^7\right)^x \cdot 2^{x-1}$$ 3. **Apply power of a power rule:** $$2^{6(2x-1)} = 2^{7x} \cdot 2^{x-1}$$ 4. **Simplify exponents:** Left side: $$2^{12x - 6}$$ Right side: $$2^{7x + x - 1} = 2^{8x - 1}$$ 5. **Since bases are equal, set exponents equal:** $$12x - 6 = 8x - 1$$ 6. **Solve for $x$:** $$12x - 8x = -1 + 6$$ $$4x = 5$$ $$x = \frac{5}{4}$$ **Final answer:** $$x = \frac{5}{4}$$