The domain of `f(x)=log|logx|` is

Options

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

Explanation

To find the domain of the function \( f(x) = \log | \log x | \), we need to ensure that the expression inside the logarithm is defined and positive. Let's go through the steps systematically.

### Step 1: Determine when \( \log x \) is defined
The logarithm function \( \log x \) is defined for \( x > 0 \). Therefore, we have:
\[ x > 0 \]

### Step 2: Determine when \( \log x \) is positive
Next, we need \( \log x \) to be greater than 0 for \( f(x) \) to be defined:
\[ \log x > 0 \]
This implies:
\[ x > e^0 \quad \text{(since } \log e^0 = 0\text{)} \]
Thus, we have:
\[ x > 1 \]

### Step 3: Determine when \( \log x \) is negative
We also need to consider when \( \log x \) is negative, as we are taking the absolute value:
\[ \log x < 0 \]
This implies:
\[ x < 1 \]
However, since we are taking the absolute value, we need to ensure that \( \log | \log x | \) is defined, which means \( \log x \) cannot be equal to 0. Therefore:
\[ \log x \neq 0 \implies x \neq 1 \]

### Step 4: Combine the conditions
From the above steps, we have two conditions:
1. \( x > 0 \)
2. \( x \neq 1 \)

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

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