1. **State the problem:** Solve for $x$ in the equation $12^{4} \times 4^{2} = 16^{x}$. 2. **Rewrite the bases in terms of prime factors:** - $12 = 2^{2} \times 3$ - $4 = 2^{2}$ - $16 = 2^{4}$ 3. **Express each term with prime factors:** $$12^{4} = (2^{2} \times 3)^{4} = 2^{8} \times 3^{4}$$ $$4^{2} = (2^{2})^{2} = 2^{4}$$ 4. **Multiply the left side:** $$12^{4} \times 4^{2} = (2^{8} \times 3^{4}) \times 2^{4} = 2^{8+4} \times 3^{4} = 2^{12} \times 3^{4}$$ 5. **Rewrite the right side:** $$16^{x} = (2^{4})^{x} = 2^{4x}$$ 6. **Set the equation:** $$2^{12} \times 3^{4} = 2^{4x}$$ 7. **Since the right side has only base 2, for equality, the factor $3^{4}$ must be 1, which is impossible unless $3^{4} = 1$. Therefore, no real $x$ satisfies this equation unless we consider the equation in terms of equality of powers of 2 only, which is not possible here.** 8. **Alternatively, if the problem intends $12^{4} \times 4^{2} = 16^{x}$ as an equality of numbers, then compute the numerical values:** - $12^{4} = 20736$ - $4^{2} = 16$ - Left side: $20736 \times 16 = 331776$ - Right side: $16^{x} = (2^{4})^{x} = 2^{4x}$ 9. **Express $331776$ as a power of 2:** - $331776 = 2^{?}$ 10. **Calculate $\log_{2}(331776)$:** $$x = \frac{\log(331776)}{\log(16)} = \frac{\log(331776)}{4 \log(2)}$$ 11. **Calculate approximate values:** - $\log(331776) \approx 5.5201$ (base 10) - $\log(2) \approx 0.3010$ 12. **Calculate $x$:** $$x = \frac{5.5201}{4 \times 0.3010} = \frac{5.5201}{1.204} \approx 4.585$$ **Final answer:** $$x \approx 4.585$$