`log_(7)log_(7)sqrt(7(sqrt(7sqrt(7))))=`

Options

A. `3log_(2)7`
B. 1-3log_(3)7`
C. `1-3log_(7)2`
D. none of these

Answer

To solve the expression \( \log_{7}\log_{7}\sqrt{7(\sqrt{7\sqrt{7}})} \), we will break it down step by step.

### Step 1: Simplify the Inner Expression
First, we need to simplify the expression inside the logarithm:
\[ \sqrt{7(\sqrt{7\sqrt{7}})} \]

### Step 2: Simplify \( \sqrt{7\sqrt{7}} \)
We know that \( \sqrt{7\sqrt{7}} = \sqrt{7 \cdot 7^{1/2}} = \sqrt{7^{3/2}} = 7^{3/4} \).

### Step 3: Substitute Back
Now substituting this back into the original expression:
\[ \sqrt{7(7^{3/4})} = \sqrt{7^{1 + 3/4}} = \sqrt{7^{7/4}} = 7^{7/8} \]

### Step 4: Rewrite the Logarithm
Now we can rewrite our original expression:
\[ \log_{7}\log_{7}(7^{7/8}) \]

### Step 5: Apply Logarithm Power Rule
Using the logarithm power rule \( \log_{a}(b^{c}) = c \cdot \log_{a}(b) \):
\[ \log_{7}(7^{7/8}) = \frac{7}{8} \cdot \log_{7}(7) \]

### Step 6: Simplify Using Logarithm Identity
Since \( \log_{7}(7) = 1 \):
\[ \log_{7}(7^{7/8}) = \frac{7}{8} \cdot 1 = \frac{7}{8} \]

### Step 7: Final Expression
Now we substitute this back into the outer logarithm:
\[ \log_{7}\left(\frac{7}{8}\right) \]

### Step 8: Use Logarithm Property
Using the property \( \log_{a}(b/c) = \log_{a}(b) - \log_{a}(c) \):
\[ \log_{7}\left(\frac{7}{8}\right) = \log_{7}(7) - \log_{7}(8) = 1 - \log_{7}(8) \]

### Step 9: Rewrite \( 8 \)
We can express \( 8 \) as \( 2^3 \):
\[ \log_{7}(8) = \log_{7}(2^3) = 3 \log_{7}(2) \]

### Step 10: Final Result
Thus, the final expression becomes:
\[ 1 - 3 \log_{7}(2) \]

### Conclusion
The final answer is:
\[ \log_{7}\log_{7}\sqrt{7(\sqrt{7\sqrt{7}})} = 1 - 3 \log_{7}(2) \]