1. **State the problem:** Solve the equation $$1616x + 9 = 3^2 (2568^{3x - 4})$$ for $x$. 2. **Rewrite the equation:** The right side is $$3^2 (2568^{3x - 4}) = 9 \times 2568^{3x - 4}$$. 3. **Isolate the exponential term:** $$1616x + 9 = 9 \times 2568^{3x - 4}$$ Divide both sides by 9: $$\frac{1616x + 9}{9} = 2568^{3x - 4}$$ Intermediate step with cancellation: $$\frac{\cancel{1616x + 9}}{\cancel{9}} = 2568^{3x - 4}$$ (Note: no common factor to cancel here, so just division) 4. **Take the natural logarithm of both sides:** $$\ln\left(\frac{1616x + 9}{9}\right) = \ln\left(2568^{3x - 4}\right)$$ 5. **Use logarithm power rule:** $$\ln\left(\frac{1616x + 9}{9}\right) = (3x - 4) \ln(2568)$$ 6. **Rewrite as:** $$\ln(1616x + 9) - \ln(9) = (3x - 4) \ln(2568)$$ 7. **Express explicitly:** $$\ln(1616x + 9) - \ln(9) = 3x \ln(2568) - 4 \ln(2568)$$ 8. **Rearrange terms:** $$\ln(1616x + 9) = 3x \ln(2568) - 4 \ln(2568) + \ln(9)$$ 9. **This is a transcendental equation in $x$ and cannot be solved algebraically in closed form.** 10. **Use numerical methods (e.g., Newton-Raphson) to approximate $x$.** **Final answer:** The solution $x$ satisfies $$\ln(1616x + 9) = 3x \ln(2568) - 4 \ln(2568) + \ln(9)$$ which can be solved numerically.