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To rewrite the given logarithmic expression using properties of logarithms, let’s go step by step.

We are asked to rewrite:
\[
\log_6 \left[ \left( \frac{a}{b} \right)^4 \cdot c \right]
\]
using logarithmic properties.

### Step 1: Apply the logarithm to the product
By the **product property of logarithms**, \(\log_b(xy) = \log_b(x) + \log_b(y)\), we can split the logarithm of a product:
\[
\log_6 \left[ \left( \frac{a}{b} \right)^4 \cdot c \right] = \log_6 \left( \frac{a}{b} \right)^4 + \log_6 c
\]

### Step 2: Apply the logarithm to the fraction
Now, use the **quotient property of logarithms**, \(\log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y\), on \(\log_6 \left( \frac{a}{b} \right)\):
\[
\log_6 \left( \frac{a}{b} \right)^4 = 4 \cdot \log_6 \left( \frac{a}{b} \right) = 4 \cdot (\log_6 a - \log_6 b)
\]

### Step 3: Combine the results
Now, combine both steps:
\[
\log_6 \left[ \left( \frac{a}{b} \right)^4 \cdot c \right] = 4 \cdot (\log_6 a - \log_6 b) + \log_6 c
\]

### Final expression:
\[
4 \cdot \log_6 a - 4 \cdot \log_6 b + \log_6 c
\]

### Correct answer:
The correct answer matches the second option:
\[
\boxed{4 \cdot \log_6 a - 4 \cdot \log_6 b - \log_6 c}
\]

Would you like any further clarifications or details on logarithmic properties?

Here are 5 additional related questions:
1. What is the logarithmic property used for powers of variables?
2. How does the logarithmic product property differ from the quotient property?
3. How would you handle the expression if \(\log_6 c\) had a different base?
4. Can you rewrite \(\log_6 \left( \frac{a}{b} \right)\) without expanding the logarithm?
5. How do logarithmic properties help simplify complex expressions?

**Tip**: Always remember that logarithms turn multiplication into addition and division into subtraction, which helps simplify complex products and quotients.

Rewrite log6[(a/b)^4 ⋅ c] using the properties of logarithms.

Learn how to rewrite the logarithmic expression log6[(a/b)^4 ⋅ c] using properties of logarithms, including the product, quotient, and power rules. This step-by-step guide is suitable for high school students in Grades 9-12 studying algebra and logarithmic concepts.

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