To express $\log_6\left[\left(\frac{a}{b}\right)^4 \cdot c\right]$ using the properties of logarithms, follow these steps:

1. Apply the logarithm to the product: [ \log_6\left[\left(\frac{a}{b}\right)^4 \cdot c\right] = \log_6\left[\left(\frac{a}{b}\right)^4\right] + \log_6(c) ]

2. Apply the logarithm to the quotient inside: $\log_6\left[\left(\frac{a}{b}\right)^4\right] = 4 \cdot \log_6\left(\frac{a}{b}\right)$

3. Break down the logarithm of the quotient: $\log_6\left(\frac{a}{b}\right) = \log_6(a) - \log_6(b)$

4. Combine everything: $\log_6\left[\left(\frac{a}{b}\right)^4 \cdot c\right] = 4\left(\log_6(a) - \log_6(b)\right) + \log_6(c)$

So, the final expression is: $\log_6\left[\left(\frac{a}{b}\right)^4 \cdot c\right] = 4\log_6(a) - 4\log_6(b) + \log_6(c)$

Would you like any further explanation or have questions on this?

Here are five related questions:

1. How can you express $\log_b(x \cdot y)$ using properties of logarithms?
2. What is the general rule for the logarithm of a power, like $\log_b(x^n)$?
3. How would you simplify $\log_6\left(a^2 \cdot b^3\right)$?
4. Can you explain how to apply the change of base formula to logarithms?
5. What happens if the base of the logarithm matches the argument, such as $\log_6(6)$?

Tip: Remember that the logarithmic properties allow you to break down complex expressions into simpler terms, which is helpful when solving logarithmic equations!