If `A=log_(2)log_(4)256+2log_(sqrt(2)).2`, then A is equal to

Options

A. `2`
B. `4`
C. `5`
D. `6`

Answer

To solve the expression \( A = \log_{2}(\log_{4}256) + 2\log_{\sqrt{2}}2 \), we will break it down step by step.

### Step 1: Simplify \( \log_{4}256 \)

We start by simplifying \( \log_{4}256 \).

\[ 256 = 4^4 \quad \text{(since } 4^4 = 256\text{)} \]

Thus,

\[ \log_{4}256 = \log_{4}(4^4) = 4 \]

### Step 2: Substitute back into the expression for \( A \)

Now, substitute \( \log_{4}256 \) back into the expression for \( A \):

\[ A = \log_{2}(4) + 2\log_{\sqrt{2}}2 \]

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

Next, we simplify \( \log_{2}(4) \):

\[ 4 = 2^2 \quad \text{(since } 2^2 = 4\text{)} \]

Thus,

\[ \log_{2}(4) = \log_{2}(2^2) = 2 \]

### Step 4: Simplify \( 2\log_{\sqrt{2}}2 \)

Now, we simplify \( 2\log_{\sqrt{2}}2 \):

Using the change of base formula, we have:

\[ \log_{\sqrt{2}}2 = \frac{\log_{2}2}{\log_{2}\sqrt{2}} \]

Since \( \sqrt{2} = 2^{1/2} \), we find:

\[ \log_{2}\sqrt{2} = \log_{2}(2^{1/2}) = \frac{1}{2} \]

Thus,

\[ \log_{\sqrt{2}}2 = \frac{1}{\frac{1}{2}} = 2 \]

So,

\[ 2\log_{\sqrt{2}}2 = 2 \times 2 = 4 \]

### Step 5: Combine the results

Now, we can combine the results from Step 3 and Step 4:

\[ A = 2 + 4 = 6 \]

### Final Answer

Thus, the value of \( A \) is:

\[ \boxed{6} \]

---