https://file.aiquickdraw.com/mathbot/file/1726125249025y6vbjtcx.jpg

The expression given in the image is:

\[
^2\log 7 \times ^3\log 16 \times ^7\log 9 = \dots
\]

This uses an unconventional notation for logarithms with superscripts. Generally, superscripts on logarithms suggest a change of base. The standard way to interpret this notation is as follows:

- \( ^2\log 7 \) means the logarithm of 7 with base 2, i.e., \( \log_2 7 \)
- \( ^3\log 16 \) means the logarithm of 16 with base 3, i.e., \( \log_3 16 \)
- \( ^7\log 9 \) means the logarithm of 9 with base 7, i.e., \( \log_7 9 \)

Thus, the expression becomes:

\[
\log_2 7 \times \log_3 16 \times \log_7 9
\]

### Step-by-step breakdown:

1. **Evaluate \( \log_2 7 \):**
   This is the logarithm of 7 in base 2. Using a calculator:
   \[
   \log_2 7 \approx 2.807
   \]

2. **Evaluate \( \log_3 16 \):**
   This is the logarithm of 16 in base 3:
   \[
   \log_3 16 \approx 2.523
   \]

3. **Evaluate \( \log_7 9 \):**
   This is the logarithm of 9 in base 7:
   \[
   \log_7 9 \approx 1.129
   \]

4. **Multiply the results:**
   Now, we multiply these values:
   \[
   2.807 \times 2.523 \times 1.129
   \]

Let me compute that for you.The result of the expression \( \log_2 7 \times \log_3 16 \times \log_7 9 \) is approximately \( 7.996 \).

Would you like more details or have any questions about this solution?

Here are five related questions to expand your understanding:
1. How can you convert a logarithm from one base to another?
2. What is the change of base formula for logarithms?
3. What are some real-world applications of logarithms?
4. How do logarithmic and exponential functions relate to each other?
5. Can you explain why logarithms with different bases can still be multiplied?

**Tip:** When calculating logarithms with unconventional bases, you can use the change of base formula, \( \log_b x = \frac{\log_c x}{\log_c b} \), where \( c \) is any base (often base 10 or base \( e \) is used).

Learn how to solve the logarithmic expression 2log7 × 3log16 × 7log9. This problem involves the change of base formula and multiplication of logarithms with different bases. Suitable for students in Grades 10-12.

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