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The expression you've provided is:

\[
4 \ln(x) - \frac{1}{3} \ln(x^2 + 1) + 2 \ln(x - 1)
\]

To simplify this logarithmic expression, we can use the logarithmic properties:

1. **Multiplication Property**: \(\ln(a) + \ln(b) = \ln(a \cdot b)\)
2. **Division Property**: \(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\)
3. **Power Property**: \(n \ln(a) = \ln(a^n)\)

### Step-by-step simplification:
1. Apply the power rule:
   \[
   4 \ln(x) = \ln(x^4), \quad 2 \ln(x - 1) = \ln((x - 1)^2)
   \]

   So, the expression becomes:
   \[
   \ln(x^4) - \frac{1}{3} \ln(x^2 + 1) + \ln((x - 1)^2)
   \]

2. Combine the logarithmic terms using the addition and subtraction properties:
   \[
   \ln\left(\frac{x^4 \cdot (x - 1)^2}{(x^2 + 1)^{1/3}}\right)
   \]

Thus, the simplified form of the expression is:
\[
\ln\left(\frac{x^4 (x - 1)^2}{(x^2 + 1)^{1/3}}\right)
\]

Would you like further clarification or details on any part of this process?

Here are 5 related questions:
1. What are the domains of logarithmic functions like \( \ln(x) \)?
2. How do the properties of logarithms apply to real-world situations?
3. Can logarithmic expressions always be simplified?
4. How do you differentiate an expression involving logarithms?
5. What happens when \( x = 1 \) in the expression \( \ln(x - 1) \)?

**Tip**: Always check the domain restrictions of the variables inside logarithms, since logarithms are only defined for positive arguments.

Simplify the expression 4 ln(x) - (1/3) ln(x^2 + 1) + 2 ln(x - 1)

This page provides a step-by-step solution to simplify the logarithmic expression 4 ln(x) - (1/3) ln(x^2 + 1) + 2 ln(x - 1) using properties of logarithms. Explore multiplication, division, and power rules applied in the solution. Suitable for high school students in Grades 10-12.

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