If `log_(sqrt(2)) sqrt(x) +log_(2)(x) + log_(4) (x^(2)) + log_(8)(x^(3)) + log_(16)(x^(4)) = 40` then x is equal to-

Options

A. 8
B. 16
C. 32
D. 256

Answer

To solve the equation

\[ \log_{\sqrt{2}} \sqrt{x} + \log_{2}(x) + \log_{4}(x^{2}) + \log_{8}(x^{3}) + \log_{16}(x^{4}) = 40, \]

we will simplify each logarithmic term step by step.

### Step 1: Simplify each logarithmic term

1. **First term:**
\[ \log_{\sqrt{2}} \sqrt{x} = \log_{2^{1/2}} x^{1/2} = \frac{1/2 \log_{2} x}{1/2} = \log_{2} x. \]

2. **Second term:**
\[ \log_{2}(x) \text{ remains as it is.} \]

3. **Third term:**
\[ \log_{4}(x^{2}) = \log_{2^{2}}(x^{2}) = \frac{2 \log_{2} x}{2} = \log_{2} x. \]

4. **Fourth term:**
\[ \log_{8}(x^{3}) = \log_{2^{3}}(x^{3}) = \frac{3 \log_{2} x}{3} = \log_{2} x. \]

5. **Fifth term:**
\[ \log_{16}(x^{4}) = \log_{2^{4}}(x^{4}) = \frac{4 \log_{2} x}{4} = \log_{2} x. \]

### Step 2: Combine the simplified terms

Now we can combine all the simplified terms:

\[ \log_{2} x + \log_{2} x + \log_{2} x + \log_{2} x + \log_{2} x = 5 \log_{2} x. \]

### Step 3: Set the equation equal to 40

Now we have:

\[ 5 \log_{2} x = 40. \]

### Step 4: Solve for \(\log_{2} x\)

Dividing both sides by 5:

\[ \log_{2} x = \frac{40}{5} = 8. \]

### Step 5: Convert logarithmic form to exponential form

Using the property of logarithms, we can convert this to:

\[ x = 2^{8}. \]

### Step 6: Calculate the value of \(x\)

Calculating \(2^{8}\):

\[ x = 256. \]

### Final Answer

Thus, the value of \(x\) is

\[ \boxed{256}. \]

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