Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume the variable is positive.)

ln*(x*2 − 1/*x*5), *x* > 1

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.

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume the variable is positive.) ln(x^2 − 1/x^5), x > 1

Learn how to expand ln((x^2 - 1) / x^5) using the properties of logarithms. This detailed step-by-step solution applies the quotient rule and power rule for logarithmic expressions, making it suitable for students in Grades 10-12.

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