1. **State the problem:** Expand the logarithmic expressions using the laws of logarithms. 2. **Recall the laws of logarithms:** - $\log_b(xy) = \log_b(x) + \log_b(y)$ - $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$ - $\log_b(x^n) = n \log_b(x)$ 3. **Part (a):** Expand $\log_{10}(x^4 y^5 z)$ Using the product rule: $$\log_{10}(x^4 y^5 z) = \log_{10}(x^4) + \log_{10}(y^5) + \log_{10}(z)$$ Using the power rule: $$= 4 \log_{10}(x) + 5 \log_{10}(y) + \log_{10}(z)$$ This matches the given expression and is correct. 4. **Part (b):** Expand $\ln \left( \frac{x^3}{\sqrt{x^2 - 36}} \right)$ with $x > 6$ Using the quotient rule: $$\ln \left( \frac{x^3}{\sqrt{x^2 - 36}} \right) = \ln(x^3) - \ln\left(\sqrt{x^2 - 36}\right)$$ Using the power rule on numerator: $$= 3 \ln(x) - \ln\left((x^2 - 36)^{1/2}\right)$$ Using the power rule on denominator: $$= 3 \ln(x) - \frac{1}{2} \ln(x^2 - 36)$$ This is the correct expanded form. 5. **Conclusion:** The correct expansions are: - (a) $4 \log_{10}(x) + 5 \log_{10}(y) + \log_{10}(z)$ - (b) $3 \ln(x) - \frac{1}{2} \ln(x^2 - 36)$ The expression in (b) given in the problem is correct, so the red 'x' is likely a mistake. Final answers: (a) $4 \log_{10}(x) + 5 \log_{10}(y) + \log_{10}(z)$ (b) $3 \ln(x) - \frac{1}{2} \ln(x^2 - 36)$