Question

If ${\log }_{x}2$ , $2^{\frac{x}{2}}$ and ${\log }_{3}x$ in G.P. ,then the value of x equals to-

• A. ${\log }_3\left({\log }_{3}2\right)$
• B. ${\log }_3\left({\log }_{2}3\right)$
• C. ${\log }_2\left({\log }_{3}2\right)$
• D. ${\log }_2\left({\log }_{2}3\right)$

Answer 1

The correct option is C ${\log }_2\left({\log }_{3}2\right)$

$lo{g}_x^2,2^{x/2}$ and $lo{g}_{3}x$ are in GP

$\left(lo{g}_{x}2\right)\left(lo{g}_{3}x\right)=(2^{x/2}{)}^2$

$\frac{log2}{logx}=\frac{logx}{log3}=2^x$

$\frac{log2}{log3}=2^x$

$lo{g}_{3}2=2^x$

$\therefore lo{g}_2\left(lo{g}_{3}2\right)=lo{g}_2\left(2^x\right)$

$\therefore x=lo{g}_2\left(lo{g}_{3}2\right)$.