1. **Problem:** Solve the equation $$\left(\frac{16}{18}\right)^{x-1} = \left(\frac{6}{9}\right)^{\frac{x}{2}}$$. 2. **Step 1: Simplify the bases.** $$\frac{16}{18} = \frac{8}{9} \quad \text{and} \quad \frac{6}{9} = \frac{2}{3}$$ 3. **Step 2: Express bases as powers of prime factors.** $$\frac{8}{9} = \frac{2^3}{3^2}$$ $$\frac{2}{3} = \frac{2^1}{3^1}$$ 4. **Step 3: Rewrite the equation using these bases:** $$\left(\frac{2^3}{3^2}\right)^{x-1} = \left(\frac{2}{3}\right)^{\frac{x}{2}}$$ 5. **Step 4: Apply power of a quotient rule:** $$\frac{2^{3(x-1)}}{3^{2(x-1)}} = \frac{2^{\frac{x}{2}}}{3^{\frac{x}{2}}}$$ 6. **Step 5: Equate the powers of 2 and 3 separately:** Since the bases are the same on both sides, the exponents must be equal: $$3(x-1) = \frac{x}{2} \quad \text{and} \quad 2(x-1) = \frac{x}{2}$$ 7. **Step 6: Solve the first equation:** $$3x - 3 = \frac{x}{2}$$ Multiply both sides by 2: $$2(3x - 3) = x$$ $$6x - 6 = x$$ Subtract $x$ from both sides: $$6x - x = 6$$ $$5x = 6$$ $$x = \frac{6}{5} = 1.2$$ 8. **Step 7: Solve the second equation:** $$2x - 2 = \frac{x}{2}$$ Multiply both sides by 2: $$4x - 4 = x$$ Subtract $x$ from both sides: $$4x - x = 4$$ $$3x = 4$$ $$x = \frac{4}{3} \approx 1.333$$ 9. **Step 8: Check for consistency.** The two values for $x$ differ, so the only way for the original equation to hold is if both sides are equal for the same $x$. Since the bases are not equal, the equation holds only if the exponents satisfy both conditions simultaneously. 10. **Step 9: Re-express the original equation using logarithms to find $x$:** Take natural logarithm on both sides: $$\ln\left(\left(\frac{8}{9}\right)^{x-1}\right) = \ln\left(\left(\frac{2}{3}\right)^{\frac{x}{2}}\right)$$ $$ (x-1) \ln\left(\frac{8}{9}\right) = \frac{x}{2} \ln\left(\frac{2}{3}\right)$$ 11. **Step 10: Solve for $x$:** $$x \ln\left(\frac{8}{9}\right) - \ln\left(\frac{8}{9}\right) = \frac{x}{2} \ln\left(\frac{2}{3}\right)$$ Bring all $x$ terms to one side: $$x \ln\left(\frac{8}{9}\right) - \frac{x}{2} \ln\left(\frac{2}{3}\right) = \ln\left(\frac{8}{9}\right)$$ Factor $x$: $$x \left(\ln\left(\frac{8}{9}\right) - \frac{1}{2} \ln\left(\frac{2}{3}\right)\right) = \ln\left(\frac{8}{9}\right)$$ 12. **Step 11: Calculate the logarithms (approximate):** $$\ln\left(\frac{8}{9}\right) \approx \ln(0.8889) \approx -0.1178$$ $$\ln\left(\frac{2}{3}\right) \approx \ln(0.6667) \approx -0.4055$$ 13. **Step 12: Substitute values:** $$x \left(-0.1178 - \frac{1}{2}(-0.4055)\right) = -0.1178$$ $$x \left(-0.1178 + 0.20275\right) = -0.1178$$ $$x (0.08495) = -0.1178$$ 14. **Step 13: Solve for $x$:** $$x = \frac{-0.1178}{0.08495} \approx -1.386$$ **Final answer:** $$\boxed{x \approx -1.386}$$