What is the value of `log_(3)2,log_(4)3.log_(5)4. . .log_(16)15` ?

Options

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

Answer

To find the value of the expression \( \log_3 2 \cdot \log_4 3 \cdot \log_5 4 \cdots \log_{16} 15 \), we will use the change of base formula for logarithms. The change of base formula states that:

\[ \log_a b = \frac{\log b}{\log a} \]

### Step-by-step Solution:

1. **Apply the Change of Base Formula**:
We can rewrite each logarithm in the product using the change of base formula:
\[ \log_3 2 = \frac{\log 2}{\log 3}, \quad \log_4 3 = \frac{\log 3}{\log 4}, \quad \log_5 4 = \frac{\log 4}{\log 5}, \ldots, \quad \log_{16} 15 = \frac{\log 15}{\log 16} \]

2. **Rewrite the Entire Expression**:
Now, substituting these into the original expression:
\[ \log_3 2 \cdot \log_4 3 \cdot \log_5 4 \cdots \log_{16} 15 = \frac{\log 2}{\log 3} \cdot \frac{\log 3}{\log 4} \cdot \frac{\log 4}{\log 5} \cdots \frac{\log 15}{\log 16} \]

3. **Observe Cancellation**:
Notice that in the product, each logarithm in the numerator cancels with the logarithm in the denominator of the next term:
\[ = \frac{\log 2}{\log 16} \]

4. **Simplify the Remaining Expression**:
We can express \( \log 16 \) in terms of \( \log 2 \):
\[ \log 16 = \log (2^4) = 4 \log 2 \]
Therefore, we have:
\[ \frac{\log 2}{\log 16} = \frac{\log 2}{4 \log 2} = \frac{1}{4} \]

5. **Final Answer**:
Thus, the value of the expression \( \log_3 2 \cdot \log_4 3 \cdots \log_{16} 15 \) is:
\[ \boxed{\frac{1}{4}} \]