Question

If ${\log }_{x}a,a^{x/2},{\log }_{b}x$ are in G.P. then $x$ is equal to

• A. ${\log }_a\left({\log }_{b}a\right)$
• B. $\frac{\log \left(\log a\right)-\log \left(\log b\right)}{\log a}$
• C. ${\log }_b\left({\log }_{a}b\right)$
• D. none of these

Answer 1

The correct option is B $\frac{\log \left(\log a\right)-\log \left(\log b\right)}{\log a}$

$lo{g}_{x}a,a^{x/2},lo{g}_{b}x.........G.P$

$(a^{x/2}{)}^2=lo{g}_{b}x\times lo{g}_{x}a$

$a^x=\frac{lo{g}_x}{lo{g}_b}\times \frac{lo{g}_a}{lo{g}_x}$

$a^x=\frac{lo{g}_a}{lo{g}_b}$

$xloga=log\left[loga\right]-log\left(logb\right)$

$x=\frac{log\left(loga\right)-log\left(logb\right)}{loga}$