1. **State the problem:** Simplify the expression $2 \log m - \log n - 4 \log p$ into a single logarithm. 2. **Recall logarithm properties:** - Power rule: $a \log b = \log b^a$ - Quotient rule: $\log a - \log b = \log \frac{a}{b}$ 3. **Apply the power rule:** $$2 \log m = \log m^2$$ $$4 \log p = \log p^4$$ 4. **Rewrite the expression:** $$2 \log m - \log n - 4 \log p = \log m^2 - \log n - \log p^4$$ 5. **Combine the logarithms using the quotient rule:** $$\log m^2 - \log n - \log p^4 = \log m^2 - \log (n p^4) = \log \frac{m^2}{n p^4}$$ **Final answer:** $$\boxed{\log \frac{m^2}{n p^4}}$$