If $l o g_{t} a, a^{t / 2}, l o g_{b} t$ be in G.P., then $t =$

l o g_{b} (l o g b) - l o g_{a} (l o g b)

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l o g_{a} (l o g a) - l o g_{a} (l o g b)

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l o g_{b} (l o g a) - l o g_{a} (l o g b)

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l o g_{a} (l o g a) - l o g_{b} (l o g b)

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Solution

The correct option is C $l o g_{a} (l o g a) - l o g_{a} (l o g b)$
As $l o g_{t} a, a^{t / 2}, l o g_{b} t$ are in G.P

$\Rightarrow (a^{\frac{t}{2}})^{2} = l o g_{t} a . l o g_{b} t = l o g_{b} a = \frac{l o g a}{l o g b}$

$a^{t} = \frac{l o g a}{l o g b}$

$\Rightarrow t = l o g_{a} (l o g_{b} a) = - l o g_{a} (l o g_{a} b) = l o g_{a} (l o g a) - l o g_{a} (l o g b)$

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