To rewrite $\log_8(x) + \log_8(8) + \log_8(y) + \log_8(12)$ in compact form, we can use the logarithmic property:

$\log_b(A) + \log_b(B) = \log_b(A \cdot B)$

Step-by-step solution:

1. Combine all terms using the addition property of logarithms: $\log_8(x) + \log_8(8) + \log_8(y) + \log_8(12) = \log_8(x \cdot 8 \cdot y \cdot 12)$

2. Simplify the product inside the logarithm: $x \cdot 8 \cdot y \cdot 12 = 96xy$

3. Rewrite the expression: $\log_8(x) + \log_8(8) + \log_8(y) + \log_8(12) = \log_8(96xy)$

Final Answer:

$\log_8(96xy)$

Would you like further details or explanations? Below are related questions to explore:

1. What are the main properties of logarithms and how are they applied?
2. How would the solution change if one term used a different logarithmic base?
3. What happens if one of the terms is subtracted instead of added?
4. How can this logarithmic expression be expanded back to its original form?
5. How do you evaluate $\log_8(96xy)$ numerically if $x$ and $y$ are known?

Tip:

Always remember to simplify expressions inside logarithms for cleaner results. For example, factorize products when possible to identify patterns easily.