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Let's break down the problem step by step to match the logarithmic expressions with their equivalent forms.

We are given the following expressions:

1. \(\ln \frac{x}{y}\)
2. \(\ln(xy)\)
3. \(\ln(x^y)\)
4. \(\frac{\ln y}{\ln x}\)

And we need to match these to the options:

A. \(\log_x y\)  
B. \(\ln x + \ln y\)  
C. \(y \ln x\)  
D. \(\ln x - \ln y\)

### Matching the expressions:

1. **\(\ln \frac{x}{y}\)**  
   Using the logarithmic property \(\ln \frac{a}{b} = \ln a - \ln b\), we get:  
   \[
   \ln \frac{x}{y} = \ln x - \ln y
   \]
   **Matches with D**.

2. **\(\ln(xy)\)**  
   Using the logarithmic property \(\ln(ab) = \ln a + \ln b\), we get:  
   \[
   \ln(xy) = \ln x + \ln y
   \]
   **Matches with B**.

3. **\(\ln(x^y)\)**  
   Using the logarithmic property \(\ln(a^b) = b \ln a\), we get:  
   \[
   \ln(x^y) = y \ln x
   \]
   **Matches with C**.

4. **\(\frac{\ln y}{\ln x}\)**  
   This matches directly with the definition of the logarithm in a different base:  
   \[
   \log_x y = \frac{\ln y}{\ln x}
   \]
   **Matches with A**.

### Final matches:
1. D  
2. B  
3. C  
4. A

Would you like further clarification on any step?

Here are five related questions:
1. What is the logarithmic property used to simplify \(\ln(ab)\)?
2. How do you convert from natural logarithms to logarithms with other bases?
3. What is the general form of the logarithmic power rule?
4. How can we express \(\log_b a\) using natural logarithms?
5. What are some applications of logarithmic identities in calculus?

**Tip**: Always remember the key logarithmic properties—product, quotient, and power rules—when simplifying expressions!

Match each of the following expressions labelled 1., 2., 3., and 4. with an equivalent expression A., B., C., and D.

Learn how to match logarithmic expressions with their equivalent forms using key properties of logarithms. This problem covers fundamental logarithmic identities and is suitable for students in Grades 11-12. Find step-by-step explanations and examples.

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