Question

If $x$ is a positive real number different from $1$ such that $log_ax, log_bx, log_cx$ are in A.P., then

• A. $b=\frac{a+c}{2}$
• B. $b=\sqrt{ac}$
• C. $c^{2}=\left(ac\right)^{log_{a}\,b}$
• D. None of (A), (B), (C) are correct

Answer 1

Answer: (C)

$2log_{b}x=log_{a}x+log_{c}x=\frac{1}{log_{x}a}+\frac{1}{log_{x}c} \Rightarrow \frac{2}{log_{x}b}=\frac{log_{x}ac}{log_{x}a\,log_{x}c}$
$\Rightarrow 2\,log_{x}c=\frac{log_{x}b}{log_{x}a}\left(log_{x}\,ac\right)=log_{a}b.log_{x}ac$
$\Rightarrow c^{2}=\left(ac\right)^{log_{a} b}$