1. **State the problem:** We want to simplify and analyze the function $$y = \ln\left(x^c (1-x)^{10} (x^3 + 1)^7\right)$$ where $c$ is a constant. 2. **Recall the logarithm property:** The logarithm of a product is the sum of the logarithms: $$\ln(abc) = \ln a + \ln b + \ln c$$ 3. **Apply the property to the function:** $$y = \ln(x^c) + \ln((1-x)^{10}) + \ln((x^3 + 1)^7)$$ 4. **Use the power rule of logarithms:** $$\ln(a^b) = b \ln a$$ 5. **Simplify each term:** $$y = c \ln x + 10 \ln(1-x) + 7 \ln(x^3 + 1)$$ 6. **Final simplified expression:** $$y = c \ln x + 10 \ln(1-x) + 7 \ln(x^3 + 1)$$ This expression is easier to analyze or differentiate if needed. **Note:** The domain restrictions are $x > 0$, $1-x > 0 \Rightarrow x < 1$, and $x^3 + 1 > 0$ which is true for all real $x$. Hence, the domain is $0 < x < 1$.