Understanding the Expression
To find the value of \( \log 5 \cdot \log_4 9 \cdot \log 2 \), we will break down each logarithmic component.

Step 1: Change of Base Formula
The logarithm \( \log_4 9 \) can be rewritten using the change of base formula:
- \( \log_4 9 = \frac{\log 9}{\log 4} \)
In this case, we can express \( \log 9 \) as \( \log 3^2 \), which simplifies to:
- \( \log 9 = 2 \log 3 \)
Now, we can express \( \log 4 \):
- \( \log 4 = \log 2^2 = 2 \log 2 \)
Putting this together, we have:
- \( \log_4 9 = \frac{2 \log 3}{2 \log 2} = \frac{\log 3}{\log 2} \)

Step 2: Substituting Back into the Original Expression
Now substitute \( \log_4 9 \) into the original expression:
- \( \log 5 \cdot \log_4 9 \cdot \log 2 = \log 5 \cdot \frac{\log 3}{\log 2} \cdot \log 2 \)
The \( \log 2 \) terms cancel out:
- \( = \log 5 \cdot \log 3 \)

Step 3: Final Calculation
We now need to find the product \( \log 5 \cdot \log 3 \). There isn't a specific numerical value unless we evaluate using a calculator or logarithm tables. However, it is a valid expression representing the product of two logarithms.

Conclusion
The simplified result of \( \log 5 \cdot \log_4 9 \cdot \log 2 \) is:
- \( \log 5 \cdot \log 3 \)
This shows the relationships between different logarithmic bases and their simplifications.