What is the value of `log_(2)(log_(3)81)`?

Options

A. 2
B. 3
C. 4
D. 9

Explanation

To solve the expression \( \log_{2}(\log_{3}81) \), we can follow these steps:

### Step 1: Simplify \( \log_{3}81 \)
First, we need to evaluate \( \log_{3}81 \). We can express 81 as a power of 3:
\[ 81 = 3^4 \]
Thus, we can rewrite the logarithm:
\[ \log_{3}81 = \log_{3}(3^4) \]
Using the property of logarithms that states \( \log_{b}(b^a) = a \), we get:
\[ \log_{3}81 = 4 \]

### Step 2: Substitute back into the original expression
Now that we have \( \log_{3}81 = 4 \), we can substitute this back into the original expression:
\[ \log_{2}(\log_{3}81) = \log_{2}(4) \]

### Step 3: Simplify \( \log_{2}(4) \)
Next, we simplify \( \log_{2}(4) \). We can express 4 as a power of 2:
\[ 4 = 2^2 \]
Thus, we can rewrite the logarithm:
\[ \log_{2}(4) = \log_{2}(2^2) \]
Again using the property of logarithms, we have:
\[ \log_{2}(4) = 2 \]

### Final Answer
Therefore, the value of \( \log_{2}(\log_{3}81) \) is:
\[ \boxed{2} \]