`f(x)=log_(e)abs(log_(e)x)`. Find the domain of f(x).

Answer

To find the domain of the function \( f(x) = \log_e |\log_e x| \), we need to determine the values of \( x \) for which this function is defined.

### Step 1: Determine the conditions for \( \log_e x \)
The logarithmic function \( \log_e x \) (or natural logarithm) is defined only for positive values of \( x \). Therefore, we have:
\[ x > 0 \]

### Step 2: Ensure \( \log_e x \) is not equal to zero
Next, we need to ensure that \( \log_e x \) is not equal to zero because we are taking the absolute value of \( \log_e x \). The logarithm equals zero when:
\[ \log_e x = 0 \implies x = e^0 = 1 \]
Thus, we need to exclude \( x = 1 \) from our domain.

### Step 3: Combine the conditions
From the above steps, we have two conditions:
1. \( x > 0 \)
2. \( x \neq 1 \)

### Step 4: Write the domain in interval notation
The values of \( x \) that satisfy these conditions are:
- From \( 0 \) to \( 1 \) (not including \( 1 \))
- From \( 1 \) to \( \infty \)

Thus, the domain of \( f(x) \) can be expressed in interval notation as:
\[ (0, 1) \cup (1, \infty) \]

### Final Answer
The domain of the function \( f(x) = \log_e |\log_e x| \) is:
\[ (0, 1) \cup (1, \infty) \]

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