te the expression as a single logarithm. \[ \frac{1}{3} \ln (x+2)^{3}+\frac{1}{2}\left[\ln (x)-\ln \left(x^{2}+3 x+2\right)^{2}\right] \]

Step 1 ：Given the expression \(\frac{1}{3} \ln (x+2)^{3}+\frac{1}{2}\left[\ln (x)-\ln \left(x^{2}+3 x+2\right)^{2}\right]\)

Step 2 ：We can use the properties of logarithms to simplify this expression. The properties of logarithms that we will use are: \(a \ln b = \ln b^a\), \(\ln a - \ln b = \ln \frac{a}{b}\), and \(\ln a + \ln b = \ln ab\)

Step 3 ：Applying these properties to the given expression, we get \(\ln x^{1/2} + \ln (x + 2)^{1/3} - \ln (x^2 + 3x + 2)^{1/2}\)

Step 4 ：Combining these logarithms into a single logarithm using the property \(\ln a + \ln b = \ln ab\) and \(\ln a - \ln b = \ln \frac{a}{b}\), we get \(\ln \left(\frac{x^{1/2}(x+2)^{1/3}}{(x^2+3x+2)^{1/2}}\right)\)

Step 5 ：Final Answer: \(\boxed{\ln \left(\frac{x^{1/2}(x+2)^{1/3}}{(x^2+3x+2)^{1/2}}\right)}\)