If `x=log_(3)5,y=log_(17)25` which one of the following is correct ?

Options

A. `xlty`
B. `x=y`
C. `xgty`
D. none of these

Answer

To solve the problem, we need to analyze the logarithmic expressions given for \( x \) and \( y \):

1. **Given**:
- \( x = \log_3 5 \)
- \( y = \log_{17} 25 \)

2. **Using the Change of Base Formula**:
The change of base formula states that:
\[ \log_a b = \frac{\log_c b}{\log_c a} \]
We can use this to express both \( x \) and \( y \) in terms of natural logarithms (or any common base). Let's use base 10 for simplicity.

- For \( x \):
\[ x = \log_3 5 = \frac{\log_{10} 5}{\log_{10} 3} \]

- For \( y \):
\[ y = \log_{17} 25 = \frac{\log_{10} 25}{\log_{10} 17} \]

3. **Expressing Logarithms**:
We know that \( 25 = 5^2 \), so we can rewrite \( y \):
\[ y = \frac{\log_{10} (5^2)}{\log_{10} 17} = \frac{2 \log_{10} 5}{\log_{10} 17} \]

4. **Comparing \( x \) and \( y \)**:
Now we have:
\[ x = \frac{\log_{10} 5}{\log_{10} 3} \]
\[ y = \frac{2 \log_{10} 5}{\log_{10} 17} \]

To compare \( x \) and \( y \), we can express them with a common term, \( \log_{10} 5 \):
- Multiply \( x \) by \( \frac{\log_{10} 17}{\log_{10} 17} \):
\[ x = \frac{\log_{10} 5 \cdot \log_{10} 17}{\log_{10} 3 \cdot \log_{10} 17} \]

- Now we can compare:
\[ x \text{ vs } y \implies \frac{\log_{10} 5 \cdot \log_{10} 17}{\log_{10} 3 \cdot \log_{10} 17} \text{ vs } \frac{2 \log_{10} 5}{\log_{10} 17} \]

- Cancel \( \log_{10} 5 \) (assuming it is positive):
\[ \frac{\log_{10} 17}{\log_{10} 3} \text{ vs } 2 \]

5. **Final Comparison**:
- We need to determine whether \( \frac{\log_{10} 17}{\log_{10} 3} \) is greater than or less than 2.
- This can be checked by calculating or estimating the logarithms:
- \( \log_{10} 17 \approx 1.230 \)
- \( \log_{10} 3 \approx 0.477 \)
- Thus:
\[ \frac{1.230}{0.477} \approx 2.58 > 2 \]

6. **Conclusion**:
Since \( \frac{\log_{10} 17}{\log_{10} 3} > 2 \), we conclude that:
\[ x < y \]

Thus, the correct relationship is \( x < y \).