Question

If $x$ is a positive real number other than $1$ such that ${\log }_{a}x,{\log }_{b}x,{\log }_{c}x$ are in A.P., then

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

Answer 1

Answer: (C)

$c^2=(ac{)}^{{\log }_{a}b}$

If ${\log }_{a}x,{\log }_{b}x,{\log }_{c}x$ are in A.P then

${\log }_{b}x=\frac{{\log }_{a}x+{\log }_{c}x}{2}$

Or

$2{\log }_{b}x={\log }_{a}x+{\log }_{c}x$

${\log }_b{x}^2=\log \left(x\right)(\frac{1}{\log a}+\frac{1}{\log c})$

$\frac{2\log x}{\log b}=\log \left(x\right)\left(\frac{\log \left(a\right)+\log \left(c\right)}{\log \left(a\right).\log \left(c\right)}\right)$

$\frac{2}{\log b}=\frac{\log \left(ac\right)}{\log a.\log c}$

$2\log c=\left(\frac{\log b}{\log a}\right)\log \left(ac\right)$

$2\log \left(c\right)={\log }_a\left(b\right)\log \left(ac\right)$

$\log \left(c^2\right)=\log \left(a{c}^{{\log }_{a}b}\right)$

$c^2={\left(ac\right)}^{{\log }_{a}b}$