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

Answer 1

The correct option is C $c^2=(ac{)}^{{\log }_{a}b}$

${\log }_{a}x,{\log }_{b}x{\log }_{c}x$ are in A.P
$\Rightarrow 2{\log }_{b}x={\log }_{a}x+{\log }_{c}x\Rightarrow \frac{2}{{\log }_{x}b}=\frac{1}{{\log }_{x}a}+\frac{1}{{\log }_{x}c}\Rightarrow \frac{2}{{\log }_{x}b}=\frac{{\log }_{x}ac}{{\log }_{x}a{\log }_{x}c}\Rightarrow 2{\log }_{x}c=\frac{{\log }_{x}b{\log }_{x}ac}{{\log }_{x}a}\Rightarrow c^2=(ac{)}^{{\log }_{a}b}$