Which of the following numbers are non positive? (A) `5^(log_(11)7)-7(log_(11)5)` (B) `log_(3)(sqrt(7)-2)` (C) `log_(7)((1)/(2))^(-1//2)` (D) `log_(sqrt(2)-1) (sqrt(2)+1)/(sqrt(2)-1)`

Options

A. `5^(log_(11)7)-7(log_(11)5)`
B. `log_(3)(sqrt(7)-2)`
C. `log_(7)((1)/(2))^(-1//2)`
D. `log_(sqrt(2)-1) (sqrt(2)+1)/(sqrt(2)-1)`

Answer

To determine which of the given numbers are non-positive, we will evaluate each option step by step.

### Option (A): \( 5^{\log_{11} 7} - 7 \log_{11} 5 \)

1. **Calculate \( \log_{11} 7 \)**:
\[ \log_{11} 7 = \frac{\log 7}{\log 11} \]
Using approximate values, \( \log 7 \approx 0.845 \) and \( \log 11 \approx 1.041 \):
\[ \log_{11} 7 \approx \frac{0.845}{1.041} \approx 0.812 \]

2. **Calculate \( 5^{\log_{11} 7} \)**:
\[ 5^{\log_{11} 7} \approx 5^{0.812} \approx 3.68 \]

3. **Calculate \( \log_{11} 5 \)**:
\[ \log_{11} 5 = \frac{\log 5}{\log 11} \]
Using approximate values, \( \log 5 \approx 0.699 \):
\[ \log_{11} 5 \approx \frac{0.699}{1.041} \approx 0.672 \]

4. **Calculate \( 7 \log_{11} 5 \)**:
\[ 7 \log_{11} 5 \approx 7 \times 0.672 \approx 4.704 \]

5. **Combine the results**:
\[ 5^{\log_{11} 7} - 7 \log_{11} 5 \approx 3.68 - 4.704 \approx -1.024 \]
Thus, option (A) is non-positive.

### Option (B): \( \log_{3}(\sqrt{7} - 2) \)

1. **Evaluate \( \sqrt{7} - 2 \)**:
\[ \sqrt{7} \approx 2.645 \quad \Rightarrow \quad \sqrt{7} - 2 \approx 0.645 \]

2. **Calculate \( \log_{3}(\sqrt{7} - 2) \)**:
Since \( \sqrt{7} - 2 > 0 \):
\[ \log_{3}(0.645) < 0 \]
Thus, option (B) is non-positive.

### Option (C): \( \log_{7}\left(\left(\frac{1}{2}\right)^{-\frac{1}{2}}\right) \)

1. **Simplify the expression**:
\[ \left(\frac{1}{2}\right)^{-\frac{1}{2}} = \sqrt{2} \]

2. **Calculate \( \log_{7}(\sqrt{2}) \)**:
Since \( \sqrt{2} < 7 \):
\[ \log_{7}(\sqrt{2}) < 1 \quad \text{(but positive)} \]
Thus, option (C) is positive.

### Option (D): \( \log_{\sqrt{2}-1}\left(\frac{\sqrt{2}+1}{\sqrt{2}-1}\right) \)

1. **Evaluate \( \frac{\sqrt{2}+1}{\sqrt{2}-1} \)**:
This can be simplified using the identity:
\[ \frac{\sqrt{2}+1}{\sqrt{2}-1} = \frac{(\sqrt{2}+1)^2}{(\sqrt{2}-1)(\sqrt{2}+1)} = \frac{3 + 2\sqrt{2}}{1} = 3 + 2\sqrt{2} \]

2. **Calculate \( \log_{\sqrt{2}-1}(3 + 2\sqrt{2}) \)**:
Since \( \sqrt{2}-1 < 1 \) and \( 3 + 2\sqrt{2} > 1 \):
The logarithm will be negative:
\[ \log_{\sqrt{2}-1}(3 + 2\sqrt{2}) < 0 \]
Thus, option (D) is non-positive.

### Conclusion:
The non-positive numbers are:
- Option (A)
- Option (B)
- Option (D)