$$ \log _{7} \log _{7} \sqrt{7(\sqrt{7 \sqrt{7}})}= $$ $$ \begin{array}{ll} ext { (a) } 3 \log _{2} 7 & ext { (b) } 1-3 \log _{3} 7 \ ext { (c) } 1-3 \log _{7} 2 & ext { (d) None of the }\end{array} $$

Question

$$ \log _{7} \log _{7} \sqrt{7(\sqrt{7 \sqrt{7}})}= $$ $$ \begin{array}{ll}	ext { (a) } 3 \log _{2} 7 & 	ext { (b) } 1-3 \log _{3} 7 \ 	ext { (c) } 1-3 \log _{7} 2 & 	ext { (d) None of the }\end{array} $$

UpStudy AI · Accepted Answer

Let's evaluate the expression step by step:

$$
\log_{7} \log_{7} \sqrt{7(\sqrt{7 \sqrt{7}})}
$$

1. **Simplify the innermost expression:**

$$
   \sqrt{7 \sqrt{7}} = \sqrt{7 \cdot 7^{1/2}} = \sqrt{7^{3/2}} = 7^{3/4}
   $$

2. **Simplify the entire expression inside the logarithms:**

$$
   \sqrt{7 \cdot \sqrt{7 \sqrt{7}}} = \sqrt{7 \cdot 7^{3/4}} = \sqrt{7^{7/4}} = 7^{7/8}
   $$

3. **Evaluate the inner logarithm:**

$$
   \log_{7}(7^{7/8}) = \frac{7}{8}
   $$

4. **Evaluate the outer logarithm:**

$$
   \log_{7}\left(\frac{7}{8}\right) = \log_{7}(7 \cdot 2^{-3}) = \log_{7}(7) + \log_{7}(2^{-3}) = 1 - 3\log_{7}(2)
   $$

Comparing this result with the given options, we find that it matches option **(c)**.

**Answer:**  
**(c) $$ 1 - 3 \log_{7} 2 $$**
The answer is $$ 1 - 3 \log_{7} 2 $$, choice (c).

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