The given problem is asking to express the logarithm of the expression $\log \left( \sqrt{\frac{x^3}{\sqrt{y}}} \right)$ in terms of $p = \log x$ and $q = \log y$.

Step-by-step solution:

1. Start with the given expression: $\log \left( \sqrt{\frac{x^3}{\sqrt{y}}} \right)$

2. Simplify the square root expression: We know that $\sqrt{A} = A^{1/2}$, so we can rewrite the expression: $\sqrt{\frac{x^3}{\sqrt{y}}} = \left( \frac{x^3}{y^{1/2}} \right)^{1/2}$

3. Use the logarithmic property $\log(a^b) = b \log(a)$: $\log \left( \left( \frac{x^3}{y^{1/2}} \right)^{1/2} \right) = \frac{1}{2} \log \left( \frac{x^3}{y^{1/2}} \right)$

4. Apply the property $\log \left( \frac{a}{b} \right) = \log a - \log b$: $\frac{1}{2} \left( \log(x^3) - \log(y^{1/2}) \right)$

5. Simplify using $\log(x^n) = n \log x$: $\frac{1}{2} \left( 3 \log x - \frac{1}{2} \log y \right)$

6. Substitute $p = \log x$ and $q = \log y$: $\frac{1}{2} \left( 3p - \frac{1}{2}q \right)$

7. Simplify the expression: $\frac{1}{2} \left( 3p - \frac{q}{2} \right) = \frac{3p}{2} - \frac{q}{4}$

Final answer:

$\log \left( \sqrt{\frac{x^3}{\sqrt{y}}} \right) = \frac{3p}{2} - \frac{q}{4}$

Would you like further clarification or details on any step?

Here are 5 related questions to explore:

1. How do logarithmic properties help simplify expressions?
2. Can you express $\log \left( x^4y^2 \right)$ in terms of $p$ and $q$?
3. What is the logarithmic expression for $\log \left( \frac{x^2}{y^3} \right)$ in terms of $p$ and $q$?
4. How do you simplify nested radicals using logarithms?
5. What other logarithmic identities are useful for solving complex expressions?

Tip: Always keep logarithmic properties in mind, such as $\log(a^b) = b \log(a)$, as they help break down complex terms into simpler components.