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The task is to estimate the value of \( \log_7 36 \) using the given logarithmic values:

- \( \log_7 6 \approx 0.9 \)
- \( \log_7 4 \approx 0.7 \)
- \( \log_7 9 \approx 1.1 \)

We can estimate \( \log_7 36 \) using the properties of logarithms, specifically the product rule:

\[
\log_b (xy) = \log_b x + \log_b y
\]

Since \( 36 = 6 \times 6 \), we can apply this property:

\[
\log_7 36 = \log_7 (6 \times 6) = \log_7 6 + \log_7 6
\]

Using the given value of \( \log_7 6 \approx 0.9 \), we find:

\[
\log_7 36 \approx 0.9 + 0.9 = 1.8
\]

Thus, the answer is approximately **1.8**.

Do you want more details or have any questions?

### Related Questions:
1. What is the change-of-base formula for logarithms?
2. How can you estimate \( \log_7 36 \) using base 10 or base 2 logarithms?
3. Can the logarithm of a product be expressed using logarithms of different bases?
4. How would you find \( \log_7 36 \) using a calculator?
5. How does the logarithmic property \( \log_b x^n = n \log_b x \) apply to this problem?

### Tip:
When you encounter logarithmic expressions involving products or powers, always check if properties like the product or power rule can simplify the calculations.

Estimate the value of log_7 36 using the given logarithmic values: log_7 6 ≈ 0.9, log_7 4 ≈ 0.7, log_7 9 ≈ 1.1.

Estimate log_7 36 using properties of logarithms and given values like log_7 6 ≈ 0.9. Learn how the product rule helps in calculating logarithmic expressions. Suitable for students in Grades 9-11.

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