We are given the expression:

$4 \log x - 4 \log (x^2 + 1) + 3 \log (x - 1)$

Step 1: Use the logarithm property $a \log b = \log b^a$

We can apply this property to the coefficients of the logarithms:

$4 \log x = \log x^4, \quad 4 \log (x^2 + 1) = \log (x^2 + 1)^4, \quad 3 \log (x - 1) = \log (x - 1)^3$

Now, the expression becomes:

$\log x^4 - \log (x^2 + 1)^4 + \log (x - 1)^3$

Step 2: Use the logarithm subtraction and addition rules

We apply the rule $\log a - \log b = \log \left( \frac{a}{b} \right)$ to the first two terms and then the rule $\log a + \log b = \log (a \cdot b)$ to combine them:

$\log \left( \frac{x^4}{(x^2 + 1)^4} \right) + \log (x - 1)^3$

This becomes:

$\log \left( \frac{x^4 (x - 1)^3}{(x^2 + 1)^4} \right)$

Final simplified expression:

$\log \left( \frac{x^4 (x - 1)^3}{(x^2 + 1)^4} \right)$

This is the simplified form of the given logarithmic expression.

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Would you like more details on any of the steps? Here are some related questions you could explore:

1. How do you simplify expressions using logarithm properties?
2. What are the main properties of logarithms for solving equations?
3. Can you provide examples of logarithm subtraction and addition rules?
4. How does the logarithmic power rule help in simplifying expressions?
5. What are the conditions under which logarithmic expressions are valid?

Tip: Always remember the domain restrictions of logarithmic functions: the arguments inside the logarithms must be positive!