1. **State the problem:** Solve for $x$ in the equation $$\log_7 11^{x-2} = 14.$$\n\n2. **Recall the logarithm power rule:** $$\log_b a^c = c \log_b a.$$ This allows us to bring the exponent down as a multiplier.\n\n3. **Apply the power rule:** $$\log_7 11^{x-2} = (x-2) \log_7 11.$$\nSo the equation becomes $$ (x-2) \log_7 11 = 14.$$\n\n4. **Isolate $x-2$:** $$x-2 = \frac{14}{\log_7 11}.$$\n\n5. **Change of base formula:** Since calculators usually compute logarithms base 10 or $e$, use $$\log_7 11 = \frac{\log 11}{\log 7}.$$\n\n6. **Substitute and simplify:** $$x-2 = \frac{14}{\frac{\log 11}{\log 7}} = 14 \times \frac{\log 7}{\log 11}.$$\n\n7. **Calculate the numerical values:** Using common logarithms (base 10), $$\log 7 \approx 0.8451, \quad \log 11 \approx 1.0414.$$\n\n8. **Evaluate:** $$x-2 = 14 \times \frac{0.8451}{1.0414} \approx 14 \times 0.8113 = 11.358.$$\n\n9. **Solve for $x$:** $$x = 11.358 + 2 = 13.358.$$\n\n10. **Round to the nearest thousandth:** $$\boxed{13.358}.$$