Arrange in ascending order
`log_(2)(x),log_(3)(x),log_(e)(x),log_(10)(x)`, if
II.`0ltxlt1`.

Answer

To arrange the logarithmic expressions \( \log_{2}(x), \log_{3}(x), \log_{e}(x), \log_{10}(x) \) in ascending order for \( 0 < x < 1 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the behavior of logarithms**:
- For any base \( a > 1 \), the logarithm \( \log_{a}(x) \) is negative when \( 0 < x < 1 \).
- The value of the logarithm decreases as the base increases.

2. **Identify the bases**:
- The bases of the logarithms in our case are \( 2, 3, e \) (approximately \( 2.718 \)), and \( 10 \).
- We know that \( 2 < e < 3 < 10 \).

3. **Comparing the logarithmic values**:
- Since \( \log_{a}(x) \) is negative and decreases as \( a \) increases, we can conclude:
- \( \log_{2}(x) < \log_{3}(x) < \log_{e}(x) < \log_{10}(x) \)
- This is because the logarithm with a smaller base (here, base 2) will yield a larger negative value than the logarithm with a larger base (here, base 10).

4. **Arranging in ascending order**:
- Therefore, we can write:
\[ \log_{10}(x) < \log_{e}(x) < \log_{3}(x) < \log_{2}(x) \]

5. **Final result**:
- The ascending order of the logarithmic expressions is:
\[ \log_{10}(x), \log_{e}(x), \log_{3}(x), \log_{2}(x) \]