To solve $\log \left( \frac{\sqrt{x}}{y} \right)$ in terms of $\log x$, $\log y$, and $\log z$, we will use the properties of logarithms, including the quotient rule, product rule, and power rule. Here are the steps:

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

2. Apply the quotient rule of logarithms: $\log \left( \frac{\sqrt{x}}{y} \right) = \log (\sqrt{x}) - \log y$

3. Simplify $\log (\sqrt{x})$ using the power rule of logarithms: $\log (\sqrt{x}) = \log (x^{1/2}) = \frac{1}{2} \log x$

4. Combine everything together: $\log \left( \frac{\sqrt{x}}{y} \right) = \frac{1}{2} \log x - \log y$

Therefore, the expression $\log \left( \frac{\sqrt{x}}{y} \right)$ in terms of $\log x$ and $\log y$ is: $\log \left( \frac{\sqrt{x}}{y} \right) = \frac{1}{2} \log x - \log y$

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:

1. How do you simplify $\log \left( \frac{a^b}{c^d} \right)$ using logarithm properties?
2. What is the logarithm property used to expand $\log (x^m \cdot y^n)$?
3. How do you simplify $\log \left( \frac{\sqrt{x}}{y^2} \right)$?
4. Can you simplify $\log (xyz)$ in terms of individual logarithms?
5. How would you approach $\log \left( \frac{x^{1/3} y^2}{z^3} \right)$?
6. What steps are involved in simplifying $\log \left( \frac{\sqrt{xy}}{z} \right)$?
7. How do you use the change of base formula to simplify logarithms?
8. What are the common mistakes to avoid when using logarithm properties?

Tip: When dealing with roots in logarithms, always remember to convert the root into a fractional exponent and then apply the power rule.