Question

помогите пожалуйста хоть до черты(((

Answer 1

1.
$36^{log_{6}(5)+log_{9}(81)}=36^{log_{6}(5)+2}=36^{log_{6}(5)+log_{6}(6^2)}=\ =36^{log_{6}(5cdot36)}=6^{2log_{6}(180)}=6^{log_{6}(108^2)}=\ =180^2=(9cdot20)^2=9^2cdot20^2=81cdot400=32400$
2.a)
$lg x-lg12=log_{0,1}(x+1)-log_{100}(4);\ x>0;x+1>0=>x>0;\ lg( frac{x}{12})= -lg(1+x)- frac{1}{2}lg(4);\ lg( frac{x}{12} )=lg((1+x)^{-1})-lg((4)^{ frac{1}{2} });\ lg( frac{x}{12} )=lg( frac{1}{2(x+1)} );\ frac{x}{6}= frac{1}{x+1}\ frac{x^2+x-6}{6(x+1)}=0\ x^2+x-6=0;\ D=1+24=25;\ x_{1}= frac{-1-5}{2}=-3notin(0;+infty);\ x_{2}= frac{-1+5}{2} =2notin(0;+infty);\ x=2;$
b)
$log_{3}^2(x-1)-2log_{ frac{1}{3}}( frac{9}{x-1} )=2^{log_{2}(7)};\ left { {{x-1>0} atop { frac{9}{x-1}>0 }} right. =>x-1>0; x>1; xin(1;+infty);\ log_{3}^2(x-1)+log_{3}(9^2)-log_{3}((x-1)^2)=7;\ log_{3}^2(x-1)-2log_{3}(x-1)=7-4=3;\ log_{3}(x-1)=t;\ t^2-2t-3=0; D=4+12=16;\ t_{1}= frac{2-4}{2}=-1=>log_{3}(x-1)=log_{3}(3^{-1})=> x-1= frac{1}{3}=>x=- frac{2}{3}\notin(1;+infty);\ t_{2}= frac{2+4}{2}=3;=>log_{3}(x-1)=log_{3}(3^3);=>x-1=27=>\ xin(0;+infty);$
x=28;

c)
$x^{ln(x)}=e^2cdot x;\ x>0;\ e^{ln(x^{ln(x)})}=e^2x;\ e^{ln^2(x)}=e^2x;\ ln(e^{ln^2(x)})=ln(e^2)+ln(x);\ ln^2(x)-ln(x)-2=0;\ D=1+8=9;\ frac{1-3}{2}=-1; x=e^{(-1)}in(0;+infty)\ frac{1+3}{2}=2; x=e^2in(0;+infty)\ x=e^{-1}; x=e^2;$

3.
a) $log_{ frac{1}{3}}(x-2)>-3log_{ frac{1}{5}} ( sqrt[3]{ frac{1}{5}} );\ x>2; xin(2;+infty);\ log_{ frac{1}{3}}(x-2)>-1;\ log_{ frac{1}{3}}(x-2)>log_{ frac{1}{3}}((frac{1}{3})^-1) ;\ x-2<3; x<5; \  xin(2;5)$
b) [tex](1 frac{11}{25})^{log_{9}(x)}>( frac{5}{6} )^{log_{ frac{1}{9}}(6-5x)} ; left { {{x>0;} atop {x< frac{6}{5} }} right.\ ( frac{36}{25} )^{log_{9(x)}}>( frac{6}{5} )^{log_{9}(6-5x)};\ 2log_{9}(x)>log_{9}(6-5x); x^2+5x-6>0;\ D=25+24=49;\ x_{1}= frac{-5-7}{2}=-6\ x_{2}= frac{-5+7}{2}=1\ left { {{01}} right. }} right.\ 1<>