The value of `log_2 3xxlog_3 2xxlog_3 4xxlog_4 3` is

Options

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

Answer

To solve the expression \( \log_2 3 \times \log_3 2 \times \log_3 4 \times \log_4 3 \), we can use the change of base formula for logarithms, which states that:

\[ \log_a b = \frac{1}{\log_b a} \]

### Step-by-Step Solution:

1. **Rewrite the logarithms using the change of base formula:**
\[ \log_2 3 = \frac{1}{\log_3 2} \]
\[ \log_4 3 = \frac{1}{\log_3 4} \]

2. **Substitute these into the original expression:**
\[ \log_2 3 \times \log_3 2 \times \log_3 4 \times \log_4 3 = \left(\frac{1}{\log_3 2}\right) \times \log_3 2 \times \log_3 4 \times \left(\frac{1}{\log_3 4}\right) \]

3. **Simplify the expression:**
- The terms \( \log_3 2 \) and \( \frac{1}{\log_3 2} \) cancel each other out.
- The terms \( \log_3 4 \) and \( \frac{1}{\log_3 4} \) also cancel each other out.

Therefore, we have:
\[ 1 \]

### Final Answer:
The value of \( \log_2 3 \times \log_3 2 \times \log_3 4 \times \log_4 3 \) is \( 1 \).