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The question is asking you to estimate the value of \( \log_8 81 \) using the provided properties of logarithms:

- \( \log_8 9 \approx 1.1 \)
- \( \log_8 12 \approx 1.2 \)
- \( \log_8 7 \approx 0.9 \)

### Approach:
Since 81 is close to \( 9^2 \), we can use the logarithmic property:
\[
\log_b (x^n) = n \cdot \log_b x
\]
In this case:
\[
\log_8 81 = \log_8 (9^2) = 2 \cdot \log_8 9
\]
Now, using the given approximation for \( \log_8 9 \):
\[
\log_8 81 \approx 2 \cdot 1.1 = 2.2
\]

### Final Answer:
The estimated value of \( \log_8 81 \) is **2.2**.

Would you like more details or have any questions?

Here are some follow-up questions:
1. What are other logarithmic properties you can use in different cases?
2. How does the change of base formula work for logarithms?
3. What happens when logarithmic bases change?
4. How would you approximate other logarithmic values using similar methods?
5. What is the relationship between exponentiation and logarithms?

**Tip:** Always use properties of logarithms like power, product, and quotient rules to simplify complex log expressions.

Estimate the value of log_8 81 using properties of logarithms and the following approximations: log_8 9 ≈ 1.1, log_8 12 ≈ 1.2, log_8 7 ≈ 0.9.

Learn how to estimate the value of log_8 81 using logarithmic properties and provided approximations. This problem covers logarithmic properties like log_b(x^n) and is suitable for students in Grades 8-10.

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