Q164 / 270 · NDA 2023 (Phase 1) · Mathematics

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

  • A
    $\log _{\mathrm{a}}\left(\log _{\mathrm{b}} \mathrm{a}\right)$
  • B
    $\log _b\left(\log _{\mathrm{a}} \mathrm{a}\right)$
  • Correct answer
    C
    $\frac{\log _a\left(\log _b a\right)}{2}$
  • D
    $\frac{\log _b\left(\log _a b\right)}{2}$

Solution

$\log x a, a x, \log b x$ are in G.P. $\left(a^x\right)^2=\log _x a \times \log _b x$ $\left(a^x\right)^2=\frac{\log a}{\log x} \times \frac{\log x}{\log b}$ $a^{2 x}=\frac{\log a}{\log b}=\log _b a$ Taking log with base $a$ $\log _a a^{2 x}=\log _a\left(\log _b a\right)$ $2 x=\log _a\left(\log _b a\right)$ $x=\frac{\log _a\left(\log _b a\right)}{2}$ 3

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

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