Question

Упростить выражение:
$sqrt{log_ab+log_ba+2}*log_{ab} a* sqrt{log^3_ab}$

Answer 1

$sqrt{log_ab+log_ba+2}cdot log_{ab} acdotsqrt{log^3_ab} = \ = sqrt{log_ab+ dfrac{1}{log_ab} +2}cdot dfrac{1}{log_{a} ab} cdot log_abcdot sqrt{log_ab} = \ =sqrt{dfrac{log^2_ab+1+2log_ab}{log_ab}}cdotdfrac{1}{log_{a} a+log_{a} b}cdotlog_abcdotsqrt{log_ab}= \ =sqrt{dfrac{(1+log_ab)^2}{log_ab}}cdot dfrac{1}{1+log_{a}b}cdot log_abcdot sqrt{log_ab}= \ =dfrac{1+log_ab}{sqrt{log_ab}}cdotdfrac{sqrt{log_ab}}{1+log_{a} b} cdot log_ab= log_ab$