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

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.

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

Explore the step-by-step simplification of the logarithmic expression ln((x-2)√x / (x^2 + 3x + 2)^2), covering logarithmic properties such as division, multiplication, and exponentiation. This problem is suitable for students in Grades 11-12.

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