1. Stated problem: Simplify and analyze the expression $$\ln\left(\frac{x^{1/2}}{x + 4}\right)$$. 2. Start by writing the logarithm of a fraction as a difference of logarithms: $$\ln\left(\frac{x^{1/2}}{x + 4}\right) = \ln\left(x^{1/2}\right) - \ln(x + 4).$$ 3. Simplify the first logarithm using the power rule of logarithms: $$\ln\left(x^{1/2}\right) = \frac{1}{2} \ln(x).$$ 4. So the expression becomes: $$\frac{1}{2} \ln(x) - \ln(x + 4).$$ 5. Domain consideration: For the expression inside the logarithms to be valid, we need: - $$x > 0$$ because of $$\ln(x)$$. - $$x + 4 > 0 \Rightarrow x > -4$$. Thus, the overall domain is $$x > 0$$ (since it is the intersection). 6. Final simplified expression is: $$\ln\left(\frac{x^{1/2}}{x + 4}\right) = \frac{1}{2} \ln(x) - \ln(x + 4).$$ This form is simpler and useful for differentiation or integration.