Question

If ${\log }_{x}a,a^{x/2},{\log }_{b}x$ are in $G.P$, then find the value of $x$.

Answer 1

${\log }_x^a,a^{x/2},{\log }_{b}x$ ...G.P.

$(a^{x/2}{)}^2=\log bx\times {\log }_{x}a$

$a^x=\frac{\log x}{\log b}\times \frac{\log a}{\log x}$

$a^x=\frac{\log a}{\log b}$

$x\log a=\log \left[\log a\right]-\log \left(\log b\right)$

$x=\frac{\log \left(\log a\right)-\log \left(\log b\right)}{\log a}$