`(log)_2(log)_2(log)_3(log)_3 27^3` is `0` b. `1` c. `2` d.`\ 3`

Options

A. 1
B. 0
C. 3
D. 2

Answer

To solve the expression \((\log_2(\log_2(\log_3(\log_3 27^3))))\), we will evaluate it step by step.

### Step 1: Evaluate \(\log_3(27^3)\)

Using the property of logarithms that states \(\log_b(a^c) = c \cdot \log_b(a)\):
\[ \log_3(27^3) = 3 \cdot \log_3(27) \]

Since \(27\) can be expressed as \(3^3\):
\[ \log_3(27) = \log_3(3^3) = 3 \]

Thus,
\[ \log_3(27^3) = 3 \cdot 3 = 9 \]

### Step 2: Evaluate \(\log_3(9)\)

Next, we need to evaluate \(\log_3(9)\):
\[ \log_3(9) = \log_3(3^2) = 2 \]

### Step 3: Evaluate \(\log_2(2)\)

Now we evaluate \(\log_2(2)\):
\[ \log_2(2) = 1 \]

### Step 4: Evaluate \(\log_2(1)\)

Finally, we evaluate \(\log_2(1)\):
\[ \log_2(1) = 0 \]

### Final Result

Putting it all together, we find:
\[ \log_2(\log_2(\log_3(\log_3(27^3)))) = \log_2(\log_2(9)) = \log_2(2) = 1 \quad \text{and} \quad \log_2(1) = 0 \]

Thus, the value of the expression is \(0\).

### Answer

The correct answer is \(0\).

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