The domain of definition of `f(x)=log_(3)|log_(e)x|`, is

Options

A. `(1,oo)`
B. `(0,oo)`
C. `(e,oo)`
D. none of these

Answer

To find the domain of the function \( f(x) = \log_3 |\log_e x| \), we need to ensure that the expression inside the logarithm is positive. Let's go through the steps to determine the domain.

### Step 1: Understand the logarithmic function
The logarithmic function \( \log_a b \) is defined only if:
1. The base \( a > 0 \) and \( a \neq 1 \).
2. The argument \( b > 0 \).

In our case, we have \( f(x) = \log_3 |\log_e x| \). Here, the base is 3, which satisfies the first condition since \( 3 > 0 \) and \( 3 \neq 1 \).

### Step 2: Set the condition for the argument of the logarithm
Next, we need to ensure that the argument \( |\log_e x| > 0 \). This means:
\[ \log_e x \neq 0 \]
and
\[ |\log_e x| > 0. \]

### Step 3: Analyze \( \log_e x \)
The expression \( \log_e x = 0 \) when \( x = 1 \). Therefore, we need to exclude \( x = 1 \) from our domain.

### Step 4: Determine when \( \log_e x \) is positive or negative
- \( \log_e x > 0 \) when \( x > 1 \).
- \( \log_e x < 0 \) when \( 0 < x < 1 \).

Thus, \( |\log_e x| > 0 \) when:
1. \( x > 1 \) (where \( \log_e x > 0 \))
2. \( 0 < x < 1 \) (where \( \log_e x < 0 \))

### Step 5: Combine the conditions
From the above analysis, we conclude:
- The domain includes \( (0, 1) \) and \( (1, \infty) \).
- However, since \( x = 1 \) must be excluded, we can write the domain as:
\[ (0, 1) \cup (1, \infty). \]

### Conclusion
The domain of the function \( f(x) = \log_3 |\log_e x| \) is:
\[ (0, 1) \cup (1, \infty). \]