`log_(5)5log_(4)9log_(3)2` simplifies to

Options

A. `2`
B. `1`
C. `5`
D. none of these

Answer

To simplify the expression \( \log_{5}(5) \log_{4}(9) \log_{3}(2) \), we will use properties of logarithms.

### Step-by-step Solution:

1. **Simplify \( \log_{5}(5) \)**:
\[ \log_{5}(5) = 1 \]
Since any logarithm of a number to its own base is 1.

2. **Simplify \( \log_{4}(9) \)**:
We can use the change of base formula:
\[ \log_{4}(9) = \frac{\log_{e}(9)}{\log_{e}(4)} \]
We can express 9 as \( 3^2 \) and 4 as \( 2^2 \):
\[ \log_{4}(9) = \frac{\log_{e}(3^2)}{\log_{e}(2^2)} = \frac{2 \log_{e}(3)}{2 \log_{e}(2)} = \frac{\log_{e}(3)}{\log_{e}(2)} \]

3. **Simplify \( \log_{3}(2) \)**:
Again, using the change of base formula:
\[ \log_{3}(2) = \frac{\log_{e}(2)}{\log_{e}(3)} \]

4. **Combine the results**:
Now substituting back into the original expression:
\[ \log_{5}(5) \log_{4}(9) \log_{3}(2) = 1 \cdot \frac{\log_{e}(3)}{\log_{e}(2)} \cdot \frac{\log_{e}(2)}{\log_{e}(3)} \]
The \( \log_{e}(3) \) and \( \log_{e}(2) \) cancel out:
\[ = 1 \]

5. **Final Result**:
Therefore, the simplified expression is:
\[ \log_{5}(5) \log_{4}(9) \log_{3}(2) = 1 \]