Statement - 1 : If `x gt 1` then `log_(10)x lt log_(3)x lt log_(e )x lt log_(2)x`.
Statement - 2 : If `0 lt x lt 1`, then `log_(x)a gt log_(x)b implies a lt b`.

Options

A. Statement - 1 is True, Statement - 2 si True , Statement -2 is a correct explanation for Statement - 1
B. Statement -1 is True, Statement -2 si True, Statemetn -2 is NOT a correct explanation for Statement -1
C. Statement - 1 is True, Statement - 2 is False
D. Statement - 1 is False, Statement - 2 is True

Explanation

To solve the given statements, we will analyze each statement step by step.

### Step 1: Analyzing Statement 1
**Statement 1**: If \( x > 1 \), then \( \log_{10} x < \log_{3} x < \log_{e} x < \log_{2} x \).

1. **Understanding Logarithmic Functions**: The logarithm function \( \log_b x \) is an increasing function when \( b > 1 \). This means that if \( b_1 < b_2 \), then \( \log_{b_1} x < \log_{b_2} x \) for \( x > 1 \).
2. **Identifying the Bases**: Here, the bases are \( 10, 3, e, \) and \( 2 \). We can order these bases: \( 10 > 3 > e > 2 \).
3. **Applying the Property**: Since \( x > 1 \), we can apply the property of logarithms:
- Since \( 10 > 3 \), we have \( \log_{10} x < \log_{3} x \).
- Since \( 3 > e \), we have \( \log_{3} x < \log_{e} x \).
- Since \( e > 2 \), we have \( \log_{e} x < \log_{2} x \).
4. **Combining the Inequalities**: Therefore, we can combine these inequalities to conclude:
\[ \log_{10} x < \log_{3} x < \log_{e} x < \log_{2} x \]

**Conclusion for Statement 1**: Statement 1 is true.

### Step 2: Analyzing Statement 2
**Statement 2**: If \( 0 < x < 1 \), then \( \log_{x} a > \log_{x} b \) implies \( a < b \).

1. **Understanding Logarithmic Inequalities**: For \( 0 < x < 1 \), the logarithmic function \( \log_{x} a \) is a decreasing function. This means that if \( \log_{x} a > \log_{x} b \), then \( a \) must be less than \( b \).
2. **Applying the Property**:
- If \( \log_{x} a > \log_{x} b \), it implies that \( a \) is positioned to the left of \( b \) on the number line when considering their logarithmic values.
- Hence, we conclude that \( a < b \).

**Conclusion for Statement 2**: Statement 2 is true.

### Final Conclusion
- **Statement 1** is true.
- **Statement 2** is also true, but it does not provide a correct explanation for Statement 1.