To expand the given logarithmic expression $\ln \left( \frac{x^2 - 1}{x^5} \right)$, we will use the properties of logarithms:

Step 1: Apply the quotient rule for logarithms

The quotient rule states that: $\ln \left( \frac{a}{b} \right) = \ln(a) - \ln(b)$ So, applying it to the expression $\ln \left( \frac{x^2 - 1}{x^5} \right)$, we get: $\ln \left( \frac{x^2 - 1}{x^5} \right) = \ln(x^2 - 1) - \ln(x^5)$

Step 2: Apply the power rule for logarithms

Next, we'll simplify the second term using the power rule, which states: $\ln(a^b) = b \cdot \ln(a)$ Thus, $\ln(x^5)$ becomes: $\ln(x^5) = 5\ln(x)$

Step 3: Write the final expanded expression

Now, substitute the simplified second term back into the expression: $\ln \left( \frac{x^2 - 1}{x^5} \right) = \ln(x^2 - 1) - 5\ln(x)$

This is the expanded form of the logarithmic expression.

Would you like more details or have any questions?

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Here are 5 related questions:

1. How do you expand $\ln \left( \frac{a^2 + b^2}{c} \right)$ using logarithmic properties?
2. What is the derivative of $\ln(x^2 - 1)$?
3. How does the logarithmic power rule apply to fractional exponents?
4. How would the expansion change if the base were not $\ln$, but $\log_b$?
5. How can you simplify $\ln(x^3 \cdot (x+1))$ using logarithm rules?

Tip: When simplifying logarithmic expressions, always apply the quotient, product, and power rules in that order to break down complex terms.