1. **State the problem:** Express $$\frac{\log m^p \sqrt{3-n}}{mn^2}$$ in terms of $$\log m$$, $$\log n$$, and $$\log p$$. 2. **Rewrite the numerator:** - $$\log m^p = p \log m$$ (power property of logarithms). - $$\log \sqrt{3-n} = \log (3-n)^{\frac{1}{2}} = \frac{1}{2} \log (3-n)$$. 3. **Rewrite the denominator:** - $$mn^2 = m \cdot n^2$$. - Using logarithm properties, $$\log(m n^2) = \log m + \log n^2 = \log m + 2 \log n$$. 4. **Express the entire fraction using log properties:** - Using the quotient property, $$\log \left(\frac{A}{B}\right) = \log A - \log B$$. 5. **Apply the logarithm:** $$\log \left(\frac{m^{p} \sqrt{3-n}}{m n^2} \right) = \log \left(m^{p} \sqrt{3-n}\right) - \log(m n^2)$$ 6. **Substitute step 2 and 3 results:** $$= p \log m + \frac{1}{2} \log (3-n) - \left(\log m + 2 \log n\right)$$ 7. **Simplify:** $$= p \log m + \frac{1}{2} \log (3-n) - \log m - 2 \log n$$ 8. **Combine like terms:** $$= (p - 1) \log m + \frac{1}{2} \log (3-n) - 2 \log n$$ Note: Since $$\log p$$ was requested but $$p$$ is an exponent (not inside a logarithm), it does not appear as $$\log p$$ in the final expression. **Final answer:** $$\boxed{(p - 1) \log m + \frac{1}{2} \log (3-n) - 2 \log n}$$