The expression you've provided is:

$\log_6 \left( \frac{x^5}{6y} \right)$

To simplify or work with this expression, we can apply logarithmic properties. Here’s a step-by-step breakdown:

Step 1: Apply the logarithm property for division:

$\log_b \left( \frac{M}{N} \right) = \log_b(M) - \log_b(N)$

So, we can split the logarithm of the fraction:

$\log_6 \left( \frac{x^5}{6y} \right) = \log_6(x^5) - \log_6(6y)$

Step 2: Simplify the first term using the logarithm power rule:

$\log_b(M^n) = n \cdot \log_b(M)$

So, $\log_6(x^5) = 5 \cdot \log_6(x)$

Step 3: Split the second term using the product rule:

$\log_b(M \cdot N) = \log_b(M) + \log_b(N)$

Thus, $\log_6(6y) = \log_6(6) + \log_6(y)$

Since $\log_6(6) = 1$ (because any logarithm of a base to itself equals 1), we get: $\log_6(6y) = 1 + \log_6(y)$

Step 4: Combine everything:

Now we can combine all the pieces: $\log_6 \left( \frac{x^5}{6y} \right) = 5 \cdot \log_6(x) - (1 + \log_6(y))$

Simplifying: $\log_6 \left( \frac{x^5}{6y} \right) = 5 \cdot \log_6(x) - 1 - \log_6(y)$

So, the simplified form is: $5 \cdot \log_6(x) - 1 - \log_6(y)$

Would you like further details or have any questions about this process?

Here are 5 related questions you might consider:

1. How does the logarithm change if the base is different?
2. Can we apply similar rules to natural logarithms ($\ln$)?
3. What happens if $x$ or $y$ is negative or zero in the logarithmic expression?
4. How do logarithmic rules apply to more complex functions, such as logarithms of sums?
5. How would you differentiate the logarithmic expression with respect to $x$ or $y$?

Tip: When simplifying logarithmic expressions, always look for opportunities to apply properties like the product rule, quotient rule, or power rule to make the expression easier to handle.