`log_(16)3.log_(17)4,log_(9)17`=_______

Options

A. `1/2`
B. `1/4`
C. `1/8`
D. `2/3`

Explanation

To solve the expression \( \log_{16} 3 \cdot \log_{17} 4 \cdot \log_{9} 17 \), we will use properties of logarithms.

### Step-by-step Solution:

1. **Use the Change of Base Formula**:
The change of base formula states that \( \log_b a = \frac{\log_k a}{\log_k b} \) for any base \( k \). We will apply this to each logarithm in the expression.
\[ \log_{16} 3 = \frac{\log 3}{\log 16}, \quad \log_{17} 4 = \frac{\log 4}{\log 17}, \quad \log_{9} 17 = \frac{\log 17}{\log 9} \]

2. **Substituting into the Expression**:
Substitute the change of base results into the original expression:
\[ \log_{16} 3 \cdot \log_{17} 4 \cdot \log_{9} 17 = \left(\frac{\log 3}{\log 16}\right) \cdot \left(\frac{\log 4}{\log 17}\right) \cdot \left(\frac{\log 17}{\log 9}\right) \]

3. **Simplifying the Expression**:
Notice that \( \log 17 \) in the numerator and denominator cancels out:
\[ = \frac{\log 3 \cdot \log 4}{\log 16 \cdot \log 9} \]

4. **Using the Property of Logarithms**:
We know that \( \log 16 = \log(4^2) = 2 \log 4 \) and \( \log 9 = \log(3^2) = 2 \log 3 \). Substitute these into the expression:
\[ = \frac{\log 3 \cdot \log 4}{(2 \log 4)(2 \log 3)} = \frac{\log 3 \cdot \log 4}{4 \log 3 \cdot \log 4} \]

5. **Final Simplification**:
The \( \log 3 \) and \( \log 4 \) terms cancel out:
\[ = \frac{1}{4} \]

### Final Answer:
\[ \log_{16} 3 \cdot \log_{17} 4 \cdot \log_{9} 17 = \frac{1}{4} \]