Let's work through the problem by applying the properties of logarithms to fully expand the given logarithmic expression:

We are given:

$\log \left( \frac{x}{\sqrt{y z^2}} \right)$

Step 1: Apply the Quotient Rule

The quotient rule states that:

$\log \left( \frac{A}{B} \right) = \log(A) - \log(B)$

Here, $A = x$ and $B = \sqrt{y z^2}$, so:

$\log \left( \frac{x}{\sqrt{y z^2}} \right) = \log(x) - \log(\sqrt{y z^2})$

Step 2: Simplify the Square Root

We know that $\sqrt{y z^2} = (y z^2)^{1/2}$, so using the power rule of logarithms, which states:

$\log(A^b) = b \log(A)$

We get:

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

Step 3: Apply the Product Rule

Now, use the product rule of logarithms on $\log(y z^2)$:

$\log(y z^2) = \log(y) + \log(z^2)$

Step 4: Apply the Power Rule Again

For $\log(z^2)$, apply the power rule again:

$\log(z^2) = 2 \log(z)$

Step 5: Combine All Steps

Now, we substitute everything back:

$\log \left( \frac{x}{\sqrt{y z^2}} \right) = \log(x) - \frac{1}{2} \left( \log(y) + 2 \log(z) \right)$

Distribute the $\frac{1}{2}$:

$\log(x) - \frac{1}{2} \log(y) - \frac{1}{2} \cdot 2 \log(z)$

Simplify:

$\log(x) - \frac{1}{2} \log(y) - \log(z)$

Final Answer:

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

This is the fully expanded form in terms of $\log x$, $\log y$, and $\log z$.

Do you need any clarifications or further details on any step? Here are some related questions you might find helpful:

1. How do you apply the product rule for logarithms?
2. Can you explain why the power rule works in logarithms?
3. What is the difference between natural logs and common logarithms?
4. How do logarithms relate to exponents?
5. What are the key properties of logarithms in simplifying expressions?

Tip: Always try to apply the logarithmic rules in small steps to avoid mistakes when expanding complex expressions!