1. **State the problem:** Simplify and analyze the function $$y=\ln\left(\frac{e^x}{e^x - 2}\right)^3$$. 2. **Recall the logarithm power rule:** $$\ln(a^b) = b \ln(a)$$. Applying this, we get: $$y = 3 \ln\left(\frac{e^x}{e^x - 2}\right)$$. 3. **Use the logarithm quotient rule:** $$\ln\left(\frac{A}{B}\right) = \ln A - \ln B$$. So, $$y = 3 \left( \ln(e^x) - \ln(e^x - 2) \right)$$. 4. **Simplify $$\ln(e^x)$$:** Since $$\ln(e^x) = x$$, the expression becomes: $$y = 3(x - \ln(e^x - 2))$$. 5. **Domain considerations:** The argument of the logarithm must be positive: $$e^x - 2 > 0 \implies e^x > 2 \implies x > \ln 2$$. 6. **Final simplified function:** $$y = 3x - 3 \ln(e^x - 2)$$ with domain $$x > \ln 2$$. This is the simplified form of the original function.