Find the domain of the function `f(x)=log_(e)(x-[x])`, where `[.]` denotes the greatest integer function.

Answer

To find the domain of the function \( f(x) = \log_e(x - [x]) \), where \([x]\) denotes the greatest integer function, we need to ensure that the argument of the logarithm is positive.

### Step-by-Step Solution:

1. **Understanding the Function**: The function is defined as \( f(x) = \log_e(x - [x]) \). Here, \([x]\) is the greatest integer less than or equal to \(x\).

2. **Condition for Logarithm**: The logarithm is defined only for positive arguments. Therefore, we need:
\[ x - [x] > 0 \]

3. **Analyzing \(x - [x]\)**: The expression \(x - [x]\) represents the fractional part of \(x\), which is denoted as \(\{x\}\). The fractional part is defined as:
\[ \{x\} = x - [x] \]
This value is always in the interval \(0 \leq \{x\} < 1\).

4. **Setting the Inequality**: From the condition \(x - [x] > 0\), we can rewrite it as:
\[ \{x\} > 0 \]
This means that \(x\) cannot be an integer because if \(x\) is an integer, \(\{x\} = 0\).

5. **Conclusion About Domain**: Therefore, the condition \( \{x\} > 0 \) implies that \(x\) must be a real number that is not an integer. Thus, the domain of the function is:
\[ \text{Domain of } f(x) = \mathbb{R} \setminus \mathbb{Z} \]
This means \(x\) can take any real number value except integers.

### Final Answer:
The domain of the function \( f(x) = \log_e(x - [x]) \) is all real numbers except integers:
\[ \text{Domain} = \{ x \in \mathbb{R} : x \text{ is not an integer} \} \]