1. **State the problem:** Simplify the expression $2\ln\left(\frac{x+2}{x+1}\right)$. 2. **Recall the logarithm power rule:** $a\ln(b) = \ln(b^a)$. This means we can move the coefficient 2 inside the logarithm as an exponent. 3. **Apply the power rule:** $$2\ln\left(\frac{x+2}{x+1}\right) = \ln\left(\left(\frac{x+2}{x+1}\right)^2\right)$$ 4. **Simplify the expression inside the logarithm:** $$\ln\left(\frac{(x+2)^2}{(x+1)^2}\right)$$ 5. **Final simplified form:** $$\boxed{\ln\left(\frac{(x+2)^2}{(x+1)^2}\right)}$$ This is the simplified form of the original expression using logarithm properties.