Domain of `log_([x+1]) (x-1)` is ( [.] represents greatest integer function)

Options

A. `(-oo, 0)`
B. `(-oo, oo)`
C. `(1, oo)`
D. `(0, oo)`

Answer

To find the domain of the function \( \log_{[x+1]}(x-1) \), where \([.]\) denotes the greatest integer function, we need to ensure that both the base and the argument of the logarithm are valid.

### Step-by-Step Solution:

1. **Identify the conditions for the logarithm**:
- The base \( [x+1] \) must be greater than 0 and not equal to 1.
- The argument \( x-1 \) must be greater than 0.

2. **Condition for the argument**:
\[ x - 1 > 0 \implies x > 1 \]

3. **Condition for the base**:
- We need \( [x+1] > 0 \):
\[ [x+1] \geq 1 \implies x + 1 \geq 1 \implies x \geq 0 \]
- We also need \( [x+1] \neq 1 \):
\[ [x+1] = 1 \implies 1 \leq x + 1 < 2 \implies 0 \leq x < 1 \]
- Therefore, we need to exclude the interval \( [0, 1) \).

4. **Combining the conditions**:
- From the argument condition, we have \( x > 1 \).
- From the base condition, we have \( x \geq 0 \) but need to exclude \( [0, 1) \).
- Thus, the valid values for \( x \) must satisfy both conditions.

5. **Final domain**:
- The only interval that satisfies both conditions is:
\[ x \in (1, \infty) \]

### Conclusion:
The domain of the function \( \log_{[x+1]}(x-1) \) is \( (1, \infty) \).

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