If `x=log_5(1000)` and `y=log_7(2058)`,then

Options

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

Answer

To solve the problem, we need to evaluate the logarithmic expressions for \( x \) and \( y \) and then compare their values.

### Step-by-Step Solution:

1. **Define the Variables:**
\[ x = \log_5(1000) \quad \text{and} \quad y = \log_7(2058) \]

2. **Convert to Base 10:**
Using the change of base formula:
\[ x = \frac{\log_{10}(1000)}{\log_{10}(5)} \quad \text{and} \quad y = \frac{\log_{10}(2058)}{\log_{10}(7)} \]

3. **Calculate \( \log_{10}(1000) \):**
Since \( 1000 = 10^3 \):
\[ \log_{10}(1000) = 3 \]
Therefore:
\[ x = \frac{3}{\log_{10}(5)} \]

4. **Approximate \( \log_{10}(5) \):**
Using a calculator or logarithm table:
\[ \log_{10}(5) \approx 0.699 \]
Thus:
\[ x \approx \frac{3}{0.699} \approx 4.29 \]

5. **Calculate \( \log_{10}(2058) \):**
Using a calculator:
\[ \log_{10}(2058) \approx 3.313 \]

6. **Approximate \( \log_{10}(7) \):**
Using a calculator:
\[ \log_{10}(7) \approx 0.845 \]
Therefore:
\[ y \approx \frac{3.313}{0.845} \approx 3.92 \]

7. **Compare \( x \) and \( y \):**
We have:
\[ x \approx 4.29 \quad \text{and} \quad y \approx 3.92 \]
Since \( y < x \), we conclude:
\[ y < x \]

### Final Conclusion:
Thus, the relation between \( x \) and \( y \) is:
\[ y < x \]