Question

If ${\log }_{x}a,a^{\frac{x}{2}}$ and ${\log }_{b}x$ are in G.P., then the value of $x$ is equal to

• A. ${\log }_a\left({\log }_{b}a\right)$
• B. ${\log }_a\left({\log }_{e}a\right)+{\log }_a\left({\log }_{e}b\right)$
• C. $-{\log }_a\left({\log }_{a}b\right)$
• D. ${\log }_a\left({\log }_{e}b\right)-{\log }_a\left({\log }_{e}a\right)$

Answer 1

The correct option is A ${\log }_a\left({\log }_{b}a\right)$

Given statement,
$\Rightarrow (a^{\frac{x}{2}}{)}^2=({\log }_{b}x)\times ({\log }_{x}a)$
$\Rightarrow a^x={\log }_{b}a$
$\Rightarrow x\log a=\log \left({\log }_{b}a\right)$
$\Rightarrow x={\log }_a\left({\log }_{b}a\right)$.