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.