Question

If ${log}_{t}a,a^{t/2},{log}_{b}t$ be in $G.P.$, then $t={log}_a\left({log}_{b}a\right)={log}_a\left(loga\right)-{log}_a\left(logb\right)$

Answer 1

Apply $y^2=zx$
$\therefore a^t={\log }_{t}a.{\log }_{b}t={\log }_{b}a$ or $\frac{\log a}{\log b}$
$\therefore t={\log }_a\left({\log }_{b}a\right)=-{\log }_a\left({\log }_{a}b\right)$$={\log }_a\left(\log a\right)-{\log }_a\left(\log b\right)$