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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Basics of Mathematics

NDA 2023 · Mathematics · Basics of Mathematics

If x a, a x and b x are in G.P, then what is x equal to?

If log⁡xa,ax\log x a, a x and log⁡bx\log b x are in G.P, then what is xx equal to?

Options

  1. Alog⁡a(log⁡ba)\log _{\mathrm{a}} \left(\log _{\mathrm{b}} \mathrm{a}\right)
  2. Blog⁡b(log⁡aa)\log _b \left(\log _{\mathrm{a}} \mathrm{a}\right)
  3. C\frac{\log _a\left(\log _b a\right)}{2}
  4. D\frac{\log _b\left(\log _a b\right)}{2}

Correct answer

C. \frac{\log _a\left(\log _b a\right)}{2}

Step-by-step solution

log⁡xa,ax,log⁡bx\log x a, a x, \log b x are in G.P. (ax)2=log⁡xa×log⁡bx(ax)2=log⁡alog⁡x×log⁡xlog⁡ba2x=log⁡alog⁡b=log⁡ba\left(a^x\right)^2=\log _x a \times \log _b x\left(a^x\right)^2=\frac{\log a}{\log x} \times \frac{\log x}{\log b}a^{2 x}=\frac{\log a}{\log b}=\log _b a Taking log with base alog⁡aa2x=log⁡aa\log _a a^{2 x}=\log _a\left(\log _b a\right)2 x=\log _a(log⁡ba)x=log⁡a(log⁡ba)2\left(\log _b a\right)x=\frac{\log _a\left(\log _b a\right)}{2} 3
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