To simplify the expression $\log_6 \left( \frac{216^x}{1296^x} \right) \log_x 6$ using logarithmic properties, follow these steps:

1. Simplify the fraction inside the logarithm: $\frac{216^x}{1296^x} = \left( \frac{216}{1296} \right)^x$

2. Simplify the base of the fraction: $\frac{216}{1296} = \frac{216 \div 216}{1296 \div 216} = \frac{1}{6}$ Therefore, $\left( \frac{216}{1296} \right)^x = \left( \frac{1}{6} \right)^x = 6^{-x}$

3. Rewrite the expression using the simplified fraction: $\log_6 \left( 6^{-x} \right) \log_x 6$

4. Apply the power rule of logarithms: $\log_6 \left( 6^{-x} \right) = -x$

5. Substitute back into the expression: $-x \log_x 6$

The simplified form of the expression is: $-x \log_x 6$

Would you like more details on this solution, or do you have any further questions?

Here are 5 questions you might want to ask next:

1. How do you convert logarithms from one base to another?
2. What are the properties of logarithms used in simplification?
3. Can you solve a logarithmic equation for a specific value of $x$?
4. How do you graph logarithmic functions?
5. What are common applications of logarithms in real-world problems?

Tip: Remember that the properties of logarithms, such as the product rule, quotient rule, and power rule, are very useful for simplifying complex logarithmic expressions.