Question

log0,1(x-2)-lgx>log0,1 3;

Answer 1

$log_{0,1}(x-2)-lgx textgreater log_{0,1} 3$
ОДЗ:
$left { {{x textgreater 0} atop {x-2 textgreater 0}} right.$
$left { {{x textgreater 0} atop {x textgreater 2}} right.$

--------------(0)------------------
                   ///////////////////
-------------------(2)-------------
                           ////////////
$x$ ∈ $(2;+$ ∞ $)$

$log_{10^{-1}}(x-2)-lgx textgreater log_{10^{-1}} 3$
$-lg(x-2)-lgx textgreater -lg 3$
$lg(x-2)+lgx textless lg 3$
$lg[x(x-2)] textless lg 3$
$lg(x^2-2x) textless lg 3$
$x^2-2x textless 3$
$x^2-2x-3 textless 0$
$D=(-2)^2-4*1*(-3)=4+12=16$
$x_1= frac{2+4}{2} =3$
$x_1= frac{2-4}{2} =-1$

-----+----(-1)---- - ----(3)------+------
                 //////////////
-------------------(2)--------------------
                          ////////////////////

Ответ: (2; 3)