`log_(2)[log_(2) {log_(2)(log_(3) 81)}]` =

Options

A. 1
B. 0
C. log 3
D. Undefined

Explanation

To solve the expression \( \log_{2}[\log_{2}(\log_{2}(\log_{3} 81))] \), we will work from the innermost logarithm to the outermost. Let's break it down step by step.

### Step 1: Calculate \( \log_{3} 81 \)

First, we need to evaluate \( \log_{3} 81 \).

We know that:
\[ 81 = 3^4 \]
Thus,
\[ \log_{3} 81 = \log_{3}(3^4) = 4 \]

### Step 2: Substitute the result into the next logarithm

Now we substitute this result into the next logarithm:
\[ \log_{2}(\log_{3} 81) = \log_{2}(4) \]

### Step 3: Calculate \( \log_{2} 4 \)

Next, we evaluate \( \log_{2} 4 \).

We know that:
\[ 4 = 2^2 \]
Thus,
\[ \log_{2} 4 = \log_{2}(2^2) = 2 \]

### Step 4: Substitute the result into the next logarithm

Now we substitute this result into the next logarithm:
\[ \log_{2}(\log_{2} 4) = \log_{2}(2) \]

### Step 5: Calculate \( \log_{2} 2 \)

Finally, we evaluate \( \log_{2} 2 \).

We know that:
\[ \log_{2} 2 = 1 \]

### Final Result

Putting it all together, we find:
\[ \log_{2}[\log_{2}(\log_{2}(\log_{3} 81))] = 1 \]

### Summary of Steps
1. \( \log_{3} 81 = 4 \)
2. \( \log_{2} 4 = 2 \)
3. \( \log_{2} 2 = 1 \)

Thus, the final answer is:
\[ \boxed{1} \]