Question:
What is the value of \(\log_7 \log_7 \sqrt{7\sqrt{7\sqrt{7}}}\) equal to?
\(\log_7 \log_7 \sqrt{7\sqrt{7\sqrt{7}}}\) का मान किसके बराबर है?
Options:
- \(3\log_7 2\)
- \(1 – 3\log_7 7\)
- \(1 – 3\log_7 2\)
- \(7/8\)
- \(3\log_7 2\)
- \(1 – 3\log_7 7\)
- \(1 – 3\log_7 2\)
- \(7/8\)
Correct Answer: c)
Explanation:
Evaluate the nested square root expression:
\[\sqrt{7\sqrt{7\sqrt{7}}} = 7^{1/2} \cdot 7^{1/4} \cdot 7^{1/8} = 7^{1/2 + 1/4 + 1/8} = 7^{7/8}\]
Now take \(\log_7\):
\[\log_7(7^{7/8}) = \frac{7}{8}\]
Now take the outer \(\log_7\):
\[\log_7\left(\frac{7}{8}\right) = \log_7 7 – \log_7 8 = 1 – \log_7(2^3) = 1 – 3\log_7 2\]
Hence, the correct option is (c).
व्यंजक का मान निकालने पर:
\[\sqrt{7\sqrt{7\sqrt{7}}} = 7^{1/2 + 1/4 + 1/8} = 7^{7/8}\]
प्रथम लघुगणक लेने पर:
\[\log_7(7^{7/8}) = \frac{7}{8}\]
पुनः \(\log_7\) लेने पर:
\[\log_7\left(\frac{7}{8}\right) = \log_7 7 – \log_7 8 = 1 – 3\log_7 2\]
अतः सही विकल्प (c) है।