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: Identify the Inner Logarithm
The function \( f(x) \) contains an inner logarithm \( \log x \). For this logarithm to be defined, \( x \) must be greater than 0.

**Hint:** Remember that the logarithm function is only defined for positive arguments.

### Step 2: Ensure the Inner Logarithm is Positive
Next, we need \( \log x \) to be positive because we are taking the logarithm of its absolute value. This means:
\[ \log x > 0 \]
This inequality implies:
\[ x > 1 \]

**Hint:** The logarithm of a number is positive when that number is greater than 1.

### Step 3: Consider the Absolute Value
Since we are dealing with \( \log | \log x | \), we need to ensure that \( \log x \) is not equal to 0, as this would make the logarithm undefined. The logarithm equals 0 when:
\[ \log x = 0 \implies x = 1 \]
Thus, \( x \) cannot be equal to 1.

**Hint:** The logarithm is zero at \( x = 1 \). We need to exclude this value from the domain.

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

Combining these, we find that the valid values for \( x \) 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 \( f(x) = \log | \log x | \) is:
\[ (0, 1) \cup (1, \infty) \]

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