The value of ${\log }_2. {\log }_3 ... {\log }_{100} {100}^{{99}^{{98}^{.{.}^{.{.}^{2^1}}}}} \mathrm{is}$

${\log }_2 . {\log }_3 ...{\log }_{99} {\log }_{100} {100}^{{99}^{{98}^{.{.}^{.2^1}}}}$

$={\log }_2 {\log }_3...{\log }_{99} {99}^{{98}^{.{.}^{.2^1}}} [{\log }_{100}100=1]$

$={\log }_2 {\log }_3 ...{\log }_{98} {98}^{{97}^{.{.}^{.{.}^{2^1}}}}$

$={\log }_2 {\log }_3 ...{\log }_{97} {97}^{{96}^{.{.}^{.2^1}}}={\log }_2 {\log }_3{3}^{2^1}$

$= {\log }_2{2}^{\text{'}} {\log }_{3}3={\log }_{2}2=1.$