Let's begin by applying the laws of logarithms to expand the given expression:

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

Step 1: Simplify the square root using the exponent rule

The square root is the same as raising the argument to the power of $\frac{1}{2}$, so we rewrite the expression as:

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

Step 2: Apply the logarithm power rule

The logarithm power rule states that $\log(a^b) = b \log(a)$. Applying this to the expression:

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

Step 3: Apply the quotient rule of logarithms

The quotient rule for logarithms states that $\log \left( \frac{a}{b} \right) = \log(a) - \log(b)$. Applying this rule to the fraction:

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

Step 4: Apply the product rule of logarithms

The product rule for logarithms states that $\log(ab) = \log(a) + \log(b)$. We apply this to the term $\log \left( x^3 (x - 1) \right)$:

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

Step 5: Simplify the logarithm of powers

The power rule of logarithms states that $\log(a^b) = b \log(a)$. Applying this rule to $\log(x^3)$:

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

Step 6: Distribute the $\frac{1}{2}$

Now, we distribute the $\frac{1}{2}$ across the terms inside the parentheses:

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

Which simplifies to:

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

Final Answer:

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

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Do you need further details or have any questions about this? Here are some related questions for deeper understanding:

1. How would you expand a logarithmic expression that involves a cube root?
2. What is the relationship between logarithms with different bases, and how can they be converted?
3. How can you simplify expressions that involve logarithms of products and quotients?
4. How can logarithmic properties be applied to solve exponential equations?
5. What is the impact of changing the argument of a logarithmic expression?

Tip: When dealing with square roots in logarithmic expressions, remember to use the exponent rule $\sqrt{a} = a^{1/2}$ to simplify the problem.