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) $\frac{\log 4}{\log 6}$.

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.