If ₓ a, a^{x} and _{b} x are in GP, then what is x equal to ?

Correct Answer is : log⁡a(log⁡ba)2{_{a}(_{b}a)}{2}2loga(logba)

Correct Answer is : log⁡a(log⁡ba)2{_{a}(_{b}a)}{2}2loga(logba)

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UPSC NDA I 2023 Mathematics Solved Paper

Question : 14 of 120

Marks: +1, -0

\log_{x} a

a^{x}

\log_{b} x

x

equal to ?

Solution:

Concept:

If three terms

p, q, r

are in GP, then

pr = q^{2}

Explanation:

Given

\log_{x} a

a^{x}

, and

\log_{b} x

are in GP.

(a^{x})^{2} = \log_{x} a \cdot \log_{b} x

Write using common logarithm:

a^{2x} = \frac{\log a}{\log x} \cdot \frac{\log x}{\log b}

Simplify:

a^{2x} = \frac{\log a}{\log b} = \log_{b} a

Take logarithm base

a

on both sides:

2x = \log_{a}(\log_{b} a)

Thus

x = \frac{\log_{a}(\log_{b} a)}{2}

Answer:

\frac{\log_{a}(\log_{b} a)}{2}

(Option C).

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