To simplify the logarithmic expression $\log_3(8^3 \cdot 9^8)$, we can apply the following logarithmic properties:

1. Product Rule: $\log_b(x \cdot y) = \log_b(x) + \log_b(y)$
2. Power Rule: $\log_b(x^n) = n \cdot \log_b(x)$

Step-by-Step Solution:

1. Apply the product rule to split the logarithm: $\log_3(8^3 \cdot 9^8) = \log_3(8^3) + \log_3(9^8)$

2. Apply the power rule to bring the exponents in front of the logarithms: $3 \cdot \log_3(8) + 8 \cdot \log_3(9)$

3. Now simplify the terms:

   • $\log_3(8)$ stays as it is.
   • $\log_3(9)$ can be rewritten because $9 = 3^2$: $\log_3(9) = \log_3(3^2) = 2 \cdot \log_3(3) = 2$

   So the expression becomes: $3 \cdot \log_3(8) + 8 \cdot 2$

4. Simplify further: $3 \cdot \log_3(8) + 16$

Thus, the simplified expression is: $3 \cdot \log_3(8) + 16$

This is the most simplified form using logarithmic properties.

Would you like any more details or further clarification? Here are 5 related questions to explore:

1. How can you simplify $\log_3(8^4 \cdot 9^6)$ using the same rules?
2. What if the base of the logarithm changes, like $\log_2(8^3 \cdot 16^2)$?
3. Can logarithms with different bases be simplified easily?
4. How do the quotient and change of base rules for logarithms work?
5. How would you handle logarithmic equations with square roots, like $\log_3(\sqrt{8} \cdot 9)$?

Tip: The power and product rules for logarithms are essential for breaking down complex expressions, especially when dealing with multiplication or exponents.