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!