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.