If _x a, ; a^ x/2 and _b x are in G.P., then x = - Vidyadip

MCQ

If ${\log _x}a,\;{a^{x/2}}$ and ${\log _b}x$ are in $G.P.$, then $x = $

A
$ - \log ({\log _b}a)$
B
$ - {\log _a}({\log _a}b)$
✓
${\log _a}({\log _e}a) - {\log _a}({\log _e}b)$
D
${\log _a}({\log _e}b) - {\log _a}({\log _e}a)$

✓

Answer

Correct option: C.

${\log _a}({\log _e}a) - {\log _a}({\log _e}b)$

c
(c) Obviously ${({a^{x/2}})^2} = {\log _x}a\;.\;{\log _b}x = {\log _b}a$

$ \Rightarrow $${a^x} = {\log _b}a$

$ \Rightarrow $$x = {\log _a}({\log _b}a)$

$ \Rightarrow $$x = {\log _a}\left( {\frac{{{{\log }_e}a}}{{{{\log }_e}b}}} \right) = {\log _a}({\log _e}a) - {\log _a}({\log _e}b)$.

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