1. The problem is to simplify the expression $\ln y = \ln \sqrt{1 + x^2} + \ln 2$ correctly. 2. Recall the logarithm property: $\ln a + \ln b = \ln (a \times b)$. 3. Applying this property, we combine the right side: $$\ln y = \ln \left(2 \times \sqrt{1 + x^2}\right)$$ 4. Exponentiate both sides to remove the logarithm: $$e^{\ln y} = e^{\ln \left(2 \times \sqrt{1 + x^2}\right)}$$ 5. Since $e^{\ln a} = a$, we get: $$y = 2 \times \sqrt{1 + x^2}$$ 6. The mistake in the original working was treating $e^{\ln a + \ln b}$ as $e^{\ln a} + e^{\ln b}$, which is incorrect. The exponential of a sum is the product of exponentials: $$e^{\ln a + \ln b} = e^{\ln a} \times e^{\ln b} = a \times b$$ 7. Therefore, the correct simplified form is: $$y = 2 \sqrt{1 + x^2}$$