To solve this problem, let's start by analyzing the given information:

Given:

• $\log x = p$
• $\log y = q$

We need to express:

$\log \left( \frac{x^3}{\sqrt{y}} \right)$

in terms of $p$ and $q$.

Step-by-Step Solution:

1. Apply the logarithm rule for division:

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

2. Apply the power rule for logarithms:

$\log(x^3) = 3 \log(x) = 3p$

$\log(\sqrt{y}) = \log(y^{1/2}) = \frac{1}{2} \log(y) = \frac{1}{2} q$

3. Substitute these values back into the original expression:

$\log \left( \frac{x^3}{\sqrt{y}} \right) = 3p - \frac{1}{2}q$

Conclusion:

The expression $\log \left( \frac{x^3}{\sqrt{y}} \right)$ in terms of $p$ and $q$ is:

$3p - \frac{1}{2}q$

Would you like more details, or do you have any questions?

Related Questions:

1. How do logarithm properties help simplify expressions?
2. Can you explain the power rule of logarithms in more detail?
3. How do you solve logarithmic equations with different bases?
4. What are some common applications of logarithms in real life?
5. How do you convert between different logarithm bases?

Tip: Remember that logarithm properties like the power rule and quotient rule are crucial for simplifying expressions!