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Question
If logxa, ax/2 and logb x are in G.P., then write the value of x.
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Solution
\[\log_x a, a^\frac{x}{2} \text { and } \log_b x \text { are in G . P } . \]
\[ \therefore \left( a^\frac{x}{2} \right)^2 = \log_x a \times \log_b x \]
\[ \Rightarrow a^x = \frac{\log_b a}{\log_b x} \times \log_b x \]
\[ \Rightarrow a^x = \log_b a \]
\[\text { Now, by taking } \log_a \text { on both the sides }: \]
\[ \Rightarrow x = \log_a \left( \log_b a \right)\]
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