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Let's solve the given logarithmic expression step-by-step. The expression is:

\[
\ln\left(\frac{\sqrt{z}}{xy}\right)
\]

### Step 1: Use logarithmic properties
We'll use the following logarithmic properties to expand the expression:

1. \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)
2. \(\ln(a^n) = n \cdot \ln(a)\)

### Step 2: Apply property 1 (quotient rule)
We can split the logarithm of the fraction into two logarithms:

\[
\ln\left(\frac{\sqrt{z}}{xy}\right) = \ln(\sqrt{z}) - \ln(xy)
\]

### Step 3: Apply property 2 (power rule)
The square root of \(z\) can be written as \(z^{1/2}\), so:

\[
\ln(\sqrt{z}) = \ln(z^{1/2}) = \frac{1}{2} \ln(z)
\]

Now, we have:

\[
\frac{1}{2} \ln(z) - \ln(xy)
\]

### Step 4: Expand \(\ln(xy)\) using property 1 again
Since \(\ln(xy)\) can be expanded as \(\ln(x) + \ln(y)\), we get:

\[
\frac{1}{2} \ln(z) - \left(\ln(x) + \ln(y)\right)
\]

### Step 5: Simplify
Distribute the negative sign:

\[
\frac{1}{2} \ln(z) - \ln(x) - \ln(y)
\]

This is the fully expanded and simplified form of the logarithmic expression.

### Final Answer:
\[
\frac{1}{2} \ln(z) - \ln(x) - \ln(y)
\]

Would you like further details or clarification?

Here are 5 related questions to explore:
1. How would the expansion change if the base were 10 instead of \(e\)?
2. How do you expand logarithms when dealing with products and powers inside the log?
3. Can you solve a logarithmic equation like this for specific values of \(z\), \(x\), and \(y\)?
4. How would you expand the logarithm if the argument were \(\frac{z^2}{x^3y}\)?
5. What is the inverse of a natural logarithm?

**Tip:** Always remember to use logarithmic rules carefully, especially when dealing with powers and products.

Learn how to expand and simplify the logarithmic expression ln(√z / xy). This step-by-step solution covers key logarithmic properties such as the quotient rule and the power rule, making it suitable for students in Grades 11-12.

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