Q151 / 270 · NDA 2018 (Phase 2) · Mathematics

What is the value of $\log _7 \log _7 \sqrt{7 \sqrt{7 \sqrt{7}}}$ equal to?

  • A
    $3 \log _2 7$
  • B
    $1-3 \log _2 7$
  • Correct answer
    C
    $1-3 \log _7 2$
  • D
    $\frac{7}{8}$

Solution

Let $\log _7 \log _7 \sqrt{7 \sqrt{7 \sqrt{7}}}=x$ then $\begin{aligned} 7^x & =\log _7 \sqrt{7 \sqrt{7 \sqrt{7}}} \\ 7^x & =\frac{1}{2} \cdot \log _7 7 \sqrt{7 \sqrt{7}} \\ 7^x & =\frac{1}{2}\left[\log _7 7+\frac{1}{2} \log _7 7 \sqrt{7}\right] \\ & =\frac{1}{2}\left[1+\frac{1}{2} \log _7 7+\frac{1}{2} \log _7 \sqrt{7}\right] \\ & =\frac{1}{2}\left[1+\frac{1}{2}+\frac{1}{4}\right] \\ 7^x & =\frac{7}{8} \\ x & =\log _7\left(\frac{7}{8}\right) \\ x & =\log _7 7-\log _7 8 \\ x & =1-3 \log _7 2 \end{aligned}$

Hence, the correct answer is option: 'C'.

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