1. **State the problem:** Simplify the expression $0.5 \log_3 16 - 6$. 2. **Recall the logarithm power rule:** $a \log_b c = \log_b c^a$. 3. Apply the power rule to $0.5 \log_3 16$: $$0.5 \log_3 16 = \log_3 16^{0.5} = \log_3 \sqrt{16}$$ 4. Simplify the square root: $$\sqrt{16} = 4$$ 5. Substitute back: $$\log_3 4 - 6$$ 6. The expression is now $\log_3 4 - 6$. This is the simplified form unless you want a decimal approximation. 7. If desired, approximate $\log_3 4$ using change of base formula: $$\log_3 4 = \frac{\ln 4}{\ln 3} \approx \frac{1.386}{1.099} \approx 1.2619$$ 8. Substitute the approximation: $$1.2619 - 6 = -4.7381$$ **Final answer:** $\log_3 4 - 6 \approx -4.7381$