Find the domain and range of the following function:
`f(x)=log_([x-1])sinx,` where [ ] denotes greatest integer function.

Explanation

To find the domain and range of the function \( f(x) = \log_{\lfloor x - 1 \rfloor} (\sin x) \), we need to analyze the conditions under which this function is defined.

### Step 1: Determine the conditions for the logarithm to be defined

1. The base of the logarithm, \( \lfloor x - 1 \rfloor \), must be greater than 0 and not equal to 1.
2. The argument of the logarithm, \( \sin x \), must be greater than 0.

### Step 2: Analyze the base of the logarithm

The base \( \lfloor x - 1 \rfloor > 0 \):
- This implies \( x - 1 \geq 1 \) or \( x \geq 2 \).

The base \( \lfloor x - 1 \rfloor \neq 1 \):
- This implies \( \lfloor x - 1 \rfloor \geq 2 \) or \( x - 1 \geq 2 \) leading to \( x \geq 3 \).

### Step 3: Analyze the sine function

The argument \( \sin x > 0 \):
- The sine function is positive in the intervals \( (0, \pi) \), \( (2\pi, 3\pi) \), \( (4\pi, 5\pi) \), and so on.
- This means \( x \) must be in these intervals.

### Step 4: Combine the conditions

Now we need to find the intersection of the intervals:
- From the base condition, \( x \geq 3 \).
- From the sine condition, \( x \) must be in intervals where \( \sin x > 0 \).

The relevant intervals for \( x \) where \( \sin x > 0 \) and \( x \geq 3 \) are:
- \( (2\pi, 3\pi) \) since \( 2\pi \approx 6.28 \) and \( 3\pi \approx 9.42 \).
- \( (4\pi, 5\pi) \) since \( 4\pi \approx 12.56 \) and \( 5\pi \approx 15.71 \).

Thus, the domain of \( f(x) \) is:
\[ \text{Domain} = [3, 2\pi) \cup (2\pi, 3\pi) \cup [4\pi, 5\pi) \cup (5\pi, \infty) \]

### Step 5: Determine the range of the function

The range of \( f(x) \):
- Since \( \sin x \) can take any value in \( (0, 1) \) when \( \sin x > 0 \), and the logarithm of a number in this interval will yield values in \( (-\infty, 0) \).
- The base \( \lfloor x - 1 \rfloor \) being greater than 1 ensures that the logarithm is defined and behaves normally.

Thus, the range of \( f(x) \) is:
\[ \text{Range} = (-\infty, 0) \]

### Final Answer

- **Domain**: \( [3, 2\pi) \cup (2\pi, 3\pi) \cup [4\pi, 5\pi) \cup (5\pi, \infty) \)
- **Range**: \( (-\infty, 0) \)