1. **State the problem:** Solve the equation $64^{2x-1} = 128^x (2^{x-1})$ for $x$. 2. **Rewrite bases as powers of 2:** - $64 = 2^6$ - $128 = 2^7$ So the equation becomes: $$ (2^6)^{2x-1} = (2^7)^x \cdot 2^{x-1} $$ 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. **Set exponents equal since bases are the same:** $$ 12x - 6 = 8x - 1 $$ 6. **Solve for $x$:** $$ 12x - 8x = -1 + 6 $$ $$ 4x = 5 $$ $$ x = \frac{5}{4} $$ **Final answer:** $$ x = \frac{5}{4} $$