To simplify the given logarithmic expression:

$\ln \left(\frac{(x-2) \cdot \sqrt{x}}{(x^2 + 3x + 2)^2}\right)$

we can apply the properties of logarithms step by step.

Step 1: Use the property of logarithms for division:

$\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)$ This gives: $\ln\left((x-2) \cdot \sqrt{x}\right) - \ln\left((x^2 + 3x + 2)^2\right)$

Step 2: Apply the logarithm property to products:

$\ln(ab) = \ln(a) + \ln(b)$ Thus: $\ln(x-2) + \ln(\sqrt{x}) - \ln\left((x^2 + 3x + 2)^2\right)$

Step 3: Simplify $\ln(\sqrt{x})$:

Recall that $\sqrt{x} = x^{1/2}$, so: $\ln(\sqrt{x}) = \frac{1}{2} \ln(x)$ Now the expression becomes: $\ln(x-2) + \frac{1}{2} \ln(x) - \ln\left((x^2 + 3x + 2)^2\right)$

Step 4: Simplify the last term:

Use the logarithm property $\ln(a^b) = b\ln(a)$: $\ln\left((x^2 + 3x + 2)^2\right) = 2\ln(x^2 + 3x + 2)$

Final Simplified Expression:

$\ln(x-2) + \frac{1}{2} \ln(x) - 2\ln(x^2 + 3x + 2)$

This is the fully simplified form of the given logarithmic expression.

Would you like further details or clarifications?

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Here are 5 related questions:

1. How does the logarithm property $\ln(a/b) = \ln(a) - \ln(b)$ work in other examples?
2. How can we differentiate the simplified logarithmic expression?
3. How does the chain rule apply to logarithmic functions in calculus?
4. What happens if $x = 2$ or $x = -1$ in this expression?
5. How can logarithms simplify complex exponential expressions?

Tip: When simplifying logarithmic expressions, always remember the key properties of logarithms for multiplication, division, and exponentiation to break down the terms systematically.