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To find the correct equivalent expression for \( \log_6 4 \), let’s go through the options and analyze each one.

We need to express \( \log_6 4 \) using the change of base formula. The change of base formula states:

\[
\log_a b = \frac{\log_c b}{\log_c a}
\]

Using this formula, we can rewrite \( \log_6 4 \) as:

\[
\log_6 4 = \frac{\log 4}{\log 6}
\]

Now let's evaluate each option:

1. **Option (A):** \( \frac{\log 6}{\log 4} \)  
   This is the reciprocal of what we want, so it is incorrect.

2. **Option (B):** \( \log 4 + \log 6 \)  
   This would be equivalent to \( \log (4 \times 6) = \log 24 \), not \( \log_6 4 \), so it is incorrect.

3. **Option (C):** \( \log 24 \)  
   This also represents \( \log (4 \times 6) \) and not \( \log_6 4 \), so it is incorrect.

4. **Option (D):** \( \frac{\log 4}{\log 6} \)  
   This matches the expression we derived using the change of base formula, so it is correct.

### Answer:
The correct answer is **Option (D)** \( \frac{\log 4}{\log 6} \).

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Would you like a further explanation or have any questions?

### Related Questions
1. How does the change of base formula work for logarithmic expressions?
2. Can \( \log_a b \) be simplified if both \( a \) and \( b \) are powers of the same number?
3. What are some applications of logarithms in real-world problems?
4. How can logarithmic properties help in solving exponential equations?
5. How do you determine the base of a logarithm if it’s not specified?

### Tip:
Always remember the change of base formula when converting between different logarithmic bases—it’s a very useful tool for simplifying and solving logarithmic expressions.

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Learn how to find the correct equivalent expression for log_6 4 by applying the change of base formula. This problem explores logarithmic properties and includes a detailed solution to clarify the concept. Suitable for high school students in Grades 10-12.

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