The value of `"log"_(sqrt(2)) sqrt(2sqrt(2sqrt(2sqrt(2))))`, is

Options

A. `(15)/(16)`
B. `(7)/(16)`
C. `(15)/(8)`
D. `(31)/(32)`

Answer

To solve the problem \( \log_{\sqrt{2}} \sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}} \), we will simplify the expression step by step.

### Step 1: Simplify the inner expression
The expression inside the logarithm is \( \sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}} \). We can denote this nested square root as \( x \):
\[ x = \sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}} \]

### Step 2: Break down the nested square roots
We can express the nested square roots in terms of powers of 2. Notice that:
\[ \sqrt{2} = 2^{1/2} \]
Thus, we can rewrite the expression:
\[ x = \sqrt{2 \cdot 2^{1/2} \cdot 2^{1/4} \cdot 2^{1/8}} \]

### Step 3: Combine the powers of 2
Now, we can combine the powers of 2:
\[ x = \sqrt{2^{1 + 1/2 + 1/4 + 1/8}} \]
Calculating the sum of the exponents:
\[ 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} = \frac{8}{8} + \frac{4}{8} + \frac{2}{8} + \frac{1}{8} = \frac{15}{8} \]
Thus, we have:
\[ x = \sqrt{2^{15/8}} = 2^{(15/8) \cdot (1/2)} = 2^{15/16} \]

### Step 4: Substitute back into the logarithm
Now we substitute \( x \) back into the logarithm:
\[ \log_{\sqrt{2}}(x) = \log_{\sqrt{2}}(2^{15/16}) \]

### Step 5: Apply the logarithm power rule
Using the logarithm property \( \log_b(a^c) = c \cdot \log_b(a) \), we can simplify:
\[ \log_{\sqrt{2}}(2^{15/16}) = \frac{15}{16} \cdot \log_{\sqrt{2}}(2) \]

### Step 6: Calculate \( \log_{\sqrt{2}}(2) \)
We know that:
\[ \log_{\sqrt{2}}(2) = 2 \quad \text{(because } \sqrt{2}^2 = 2\text{)} \]

### Step 7: Final calculation
Now we can substitute this back into our equation:
\[ \log_{\sqrt{2}}(2^{15/16}) = \frac{15}{16} \cdot 2 = \frac{30}{16} = \frac{15}{8} \]

### Conclusion
Thus, the value of \( \log_{\sqrt{2}} \sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}} \) is:
\[ \frac{15}{8} \]