1. **State the problem:** We need to simplify and understand the function $g(x) = \ln\left(x\sqrt{x^2 - 1}\right)$. 2. **Recall the logarithm property:** $\ln(ab) = \ln a + \ln b$. This allows us to split the logarithm of a product into a sum of logarithms. 3. **Apply the property:** $$g(x) = \ln x + \ln \sqrt{x^2 - 1}$$ 4. **Simplify the square root inside the logarithm:** Recall that $\sqrt{y} = y^{1/2}$, so $$\ln \sqrt{x^2 - 1} = \ln (x^2 - 1)^{1/2} = \frac{1}{2} \ln (x^2 - 1)$$ 5. **Rewrite the function:** $$g(x) = \ln x + \frac{1}{2} \ln (x^2 - 1)$$ 6. **Domain considerations:** For $g(x)$ to be defined, the arguments of the logarithms must be positive: - $x > 0$ - $x^2 - 1 > 0 \Rightarrow |x| > 1$ Combining these, the domain is $x > 1$. **Final simplified form:** $$g(x) = \ln x + \frac{1}{2} \ln (x^2 - 1), \quad x > 1$$