If `x = "log"_(2) 3 " and " y = "log"_(1//2) 5`, then

Options

A. `x gt y`
B. `x lt y`
C. `x = y`
D. none of these

Answer

To solve the problem, we need to compare the values of \( x \) and \( y \) given as follows:

1. \( x = \log_2 3 \)
2. \( y = \log_{1/2} 5 \)

### Step 1: Rewrite \( y \)
We can rewrite \( y \) using the change of base property of logarithms. The base \( \frac{1}{2} \) can be expressed as \( 2^{-1} \).

Using the property \( \log_a b = \frac{\log_c b}{\log_c a} \), we can express \( y \) as:
\[ y = \log_{1/2} 5 = \log_{2^{-1}} 5 = \frac{\log_2 5}{\log_2 (2^{-1})} \]
Since \( \log_2 (2^{-1}) = -1 \), we have:
\[ y = -\log_2 5 \]

### Step 2: Compare \( x \) and \( y \)
Now we have:
- \( x = \log_2 3 \)
- \( y = -\log_2 5 \)

We can rewrite \( y \) as:
\[ y = \log_2 (5^{-1}) = \log_2 \left(\frac{1}{5}\right) \]

### Step 3: Compare the logarithmic values
Now we need to compare \( x \) and \( y \):
- \( x = \log_2 3 \)
- \( y = \log_2 \left(\frac{1}{5}\right) \)

Since both logarithms have the same base (base 2), we can directly compare the arguments:
- \( 3 \) and \( \frac{1}{5} \)

Since \( 3 > \frac{1}{5} \), we can conclude that:
\[ \log_2 3 > \log_2 \left(\frac{1}{5}\right) \]
Thus, \( x > y \).

### Final Conclusion
The final result is:
\[ x > y \]