This is similar to Try It \#4 in the OpenStax text. Expand \( \ln \left(\frac{1}{x^{3}}\right) \)

To **expand** the natural logarithm expression \( \ln\left(\frac{1}{x^{3}}\right) \), you can apply logarithmic properties step by step. Here's how you can do it: ### Step 1: Apply the Quotient Rule for Logarithms The quotient rule states that: \[ \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \] Apply this to the given expression: \[ \ln\left(\frac{1}{x^{3}}\right) = \ln(1) - \ln(x^{3}) \] ### Step 2: Simplify \(\ln(1)\) Recall that: \[ \ln(1) = 0 \] So the expression becomes: \[ 0 - \ln(x^{3}) = -\ln(x^{3}) \] ### Step 3: Apply the Power Rule for Logarithms The power rule states that: \[ \ln(a^{b}) = b \cdot \ln(a) \] Apply this to \( \ln(x^{3}) \): \[ \ln(x^{3}) = 3 \cdot \ln(x) \] ### Step 4: Combine the Results Substitute back into the expression: \[ -\ln(x^{3}) = -3 \cdot \ln(x) \] ### Final Expanded Form \[ \ln\left(\frac{1}{x^{3}}\right) = -3 \ln(x) \] **Conclusion:** The expanded form of \( \ln\left(\frac{1}{x^{3}}\right) \) is: \[ -3 \ln(x) \]