Question

If \(x\) is a positive real number different from 1 such that \(\log _{a} x, \log _{b} x, \log _{c} x\) are in A.P., then
(A) \(b=\frac{a+c}{2}\)
(B) \(b=\sqrt{a c}\)
(C) \(\mathrm{c}^{2}=(\mathrm{ac})^{\log \mathrm{b} \mathrm{b}}\)
(D) None of (A), (B), (C) are correct

Answer 1

Ans: (C)
$$
\begin{aligned}
&\text { Hint : } 2 \log _{b} x=\log _{a} x+\log _{c} x=\frac{1}{\log _{x} a}+\frac{1}{\log _{x} c} \Rightarrow \frac{2}{\log _{x} b}=\frac{\log _{x} a c}{\log _{x} a \log _{x} c} \Rightarrow 2 \log _{x} c=\frac{\log _{x} b}{\log _{x} a}\left(\log _{x} a c\right)=\log _{a} b \cdot \log _{x} a c \\
&\Rightarrow c^{2}=(a c)^{\log _{a} b}
\end{aligned}
$$

Answer 2

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