We can simplify the expression $6 \log_6(36)$ using the properties of logarithms.

Step 1: Apply the power rule for logarithms

The power rule states that: $a \log_b(x) = \log_b(x^a)$ So, we can rewrite the expression as: $6 \log_6(36) = \log_6(36^6)$

Step 2: Simplify $36^6$

First, simplify $36^6$. We know that: $36 = 6^2$ Thus, $36^6 = (6^2)^6 = 6^{12}$.

Step 3: Apply the logarithmic identity

Now that we have $\log_6(6^{12})$, we use the identity: $\log_b(b^a) = a$ In this case, $\log_6(6^{12}) = 12$.

Thus, the simplified expression is: $6 \log_6(36) = 12$

Final Answer:

$6 \log_6(36) = 12$

Do you have any questions or would you like further details? Here are 5 related questions to deepen understanding:

1. How does the power rule of logarithms work for different bases?
2. Can you apply the change of base formula to the same problem?
3. How would the answer change if the base of the logarithm were different, say $\log_2(36)$?
4. What happens if you apply the logarithmic identity $\log_b(b^a) = a$ incorrectly?
5. How can logarithms be used to solve exponential equations?

Tip: Always check if the base of the logarithm and the number inside the logarithm can be related as powers. This can help simplify logarithmic expressions faster.