The domain of definition of `f(x) = log_(2) (log_(3) (log_(4) x))`, is

Options

A. `[4, oo)`
B. `(4, oo)`
C. `(-oo, 4)`
D. none of these

Explanation

To find the domain of the function \( f(x) = \log_{2}(\log_{3}(\log_{4} x)) \), we need to ensure that all logarithmic expressions are defined and positive. Let's go through the steps systematically.

### Step 1: Determine the innermost logarithm
The innermost function is \( \log_{4} x \). For this logarithm to be defined, the argument \( x \) must be greater than 0:
\[ x > 0 \]
Additionally, for \( \log_{4} x \) to be positive, we need:
\[ \log_{4} x > 0 \]
This implies:
\[ x > 4^0 = 1 \]
So, from this step, we have:
\[ x > 4 \]

### Step 2: Move to the next logarithm
Next, we consider \( \log_{3}(\log_{4} x) \). For this logarithm to be defined and positive, we require:
\[ \log_{4} x > 0 \]
From Step 1, we already established that \( x > 4 \), which ensures \( \log_{4} x > 0 \).

### Step 3: Consider the outermost logarithm
Now we look at the outermost function \( \log_{2}(\log_{3}(\log_{4} x)) \). For this logarithm to be defined and positive, we need:
\[ \log_{3}(\log_{4} x) > 0 \]
This implies:
\[ \log_{4} x > 3^0 = 1 \]
Now, we need to solve:
\[ \log_{4} x > 1 \]
This can be rewritten as:
\[ x > 4^1 = 4 \]

### Conclusion
Combining all the conditions, we find that the only relevant condition is \( x > 4 \). Therefore, the domain of the function \( f(x) \) is:
\[ (4, \infty) \]

### Final Answer
The domain of definition of \( f(x) = \log_{2}(\log_{3}(\log_{4} x)) \) is \( (4, \infty) \).