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.