Question

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

Answer 1

$lo{g}_{x}2,2^{\frac{x}{2}},lo{g}_{3}x$ are in GP.

$\frac{lo{g}_x^2}{2^{\frac{x}{2}}}=\frac{1}{r}=\frac{2^{\frac{x}{2}}}{lo{g}_3^x}$

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

$\Rightarrow 2^x=lo{g}_3^2$

$\Rightarrow x=lo{g}_2\left(lo{g}_3^2\right)$

$\Rightarrow x=\frac{log\left(lo{g}_3^2\right)}{log2}$