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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!

Expand the logarithm fully using the properties of logs. Express the final answer in terms of log x, log y, and log z.

log(x / √(y * z^2))

Learn how to expand the logarithmic expression log(x / √(y * z^2)) fully using logarithm properties. This step-by-step guide covers key logarithmic concepts such as the quotient rule, product rule, and power rule, suitable for students in Grades 10-12.

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