The expression from the image you provided is:

$2 \left[ 3 \ln{x} - \ln{(x + 1)} - \ln{(x - 1)} \right]$

Let’s simplify this step by step.

Step 1: Simplify the terms inside the brackets

We can combine the logarithmic terms using logarithmic properties. Specifically, the property:

$\ln{a} - \ln{b} = \ln{\left(\frac{a}{b}\right)}$

Apply this to the second and third terms:

$3 \ln{x} - \ln{(x + 1)} - \ln{(x - 1)} = 3 \ln{x} - \ln{\left( (x+1)(x-1) \right)}$

Step 2: Simplify the product inside the logarithm

We simplify the product inside the logarithm:

$(x+1)(x-1) = x^2 - 1$

Thus, the expression becomes:

$3 \ln{x} - \ln{(x^2 - 1)}$

Step 3: Apply logarithmic properties again

Now apply the logarithmic property:

$a \ln{b} = \ln{(b^a)}$

to the first term $3 \ln{x}$:

$3 \ln{x} = \ln{(x^3)}$

Thus, the expression inside the brackets is:

$\ln{(x^3)} - \ln{(x^2 - 1)}$

Step 4: Combine the logarithms

Using the same logarithmic subtraction property again:

$\ln{a} - \ln{b} = \ln{\left(\frac{a}{b}\right)}$

We get:

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

Step 5: Multiply by the factor outside the brackets

Finally, we multiply by 2, as indicated by the factor outside the brackets:

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

Using the property $a \ln{b} = \ln{(b^a)}$, we can write the final expression as:

$\ln{\left(\frac{x^3}{x^2 - 1}\right)^2} = \ln{\left(\frac{x^6}{(x^2 - 1)^2}\right)}$

Final Simplified Expression:

$\ln{\left(\frac{x^6}{(x^2 - 1)^2}\right)}$

Would you like further details on any step or have any questions?

Here are 5 related questions:

1. How can logarithmic identities be used in more complex expressions?
2. What are the properties of logarithms that help in solving equations?
3. How can we use logarithms to solve exponential equations?
4. What are the applications of natural logarithms in real-world problems?
5. How do logarithmic transformations help in graphing non-linear data?

Tip: When simplifying logarithmic expressions, always look for opportunities to apply properties like product, quotient, and power rules to combine or reduce terms efficiently.