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

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Final Answer: $\ln (\frac{x^{1 / 2} (x + 2)^{1 / 3}}{(x^{2} + 3 x + 2)^{1 / 2}})$

Steps

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

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 $\ln a + \ln b = \ln a b$

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

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

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

te the expression as a single logarithm.13ln⁡(x+2)3+12[ln⁡(x)−ln⁡(x2+3x+2)2]

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