Предмет: Алгебра,
автор: linkactlink
помогите пожалуйста хоть до черты(((
Приложения:
Ответы
Автор ответа:
0
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](https://tex.z-dn.net/?f=36%5E%7Blog_%7B6%7D%285%29%2Blog_%7B9%7D%2881%29%7D%3D36%5E%7Blog_%7B6%7D%285%29%2B2%7D%3D36%5E%7Blog_%7B6%7D%285%29%2Blog_%7B6%7D%286%5E2%29%7D%3D%5C%0A%3D36%5E%7Blog_%7B6%7D%285cdot36%29%7D%3D6%5E%7B2log_%7B6%7D%28180%29%7D%3D6%5E%7Blog_%7B6%7D%28108%5E2%29%7D%3D%5C%0A%3D180%5E2%3D%289cdot20%29%5E2%3D9%5E2cdot20%5E2%3D81cdot400%3D32400 "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;](https://tex.z-dn.net/?f=lg+x-lg12%3Dlog_%7B0%2C1%7D%28x%2B1%29-log_%7B100%7D%284%29%3B%5C%0Ax%26gt%3B0%3Bx%2B1%26gt%3B0%3D%26gt%3Bx%26gt%3B0%3B%5C%0Alg%28+frac%7Bx%7D%7B12%7D%29%3D+-lg%281%2Bx%29-+frac%7B1%7D%7B2%7Dlg%284%29%3B%5C%0Alg%28++frac%7Bx%7D%7B12%7D++%29%3Dlg%28%281%2Bx%29%5E%7B-1%7D%29-lg%28%284%29%5E%7B+frac%7B1%7D%7B2%7D+%7D%29%3B%5C%0Alg%28+frac%7Bx%7D%7B12%7D+%29%3Dlg%28+frac%7B1%7D%7B2%28x%2B1%29%7D+%29%3B%5C%0A+frac%7Bx%7D%7B6%7D%3D++frac%7B1%7D%7Bx%2B1%7D%5C%0A+++++frac%7Bx%5E2%2Bx-6%7D%7B6%28x%2B1%29%7D%3D0%5C%0Ax%5E2%2Bx-6%3D0%3B%5C%0A+D%3D1%2B24%3D25%3B%5C%0Ax_%7B1%7D%3D+frac%7B-1-5%7D%7B2%7D%3D-3notin%280%3B%2Binfty%29%3B%5C%0Ax_%7B2%7D%3D+frac%7B-1%2B5%7D%7B2%7D+%3D2notin%280%3B%2Binfty%29%3B%5C%0Ax%3D2%3B "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);\](https://tex.z-dn.net/?f=log_%7B3%7D%5E2%28x-1%29-2log_%7B+frac%7B1%7D%7B3%7D%7D%28+frac%7B9%7D%7Bx-1%7D+%29%3D2%5E%7Blog_%7B2%7D%287%29%7D%3B%5C%0A+left+%7B+%7B%7Bx-1%26gt%3B0%7D+atop+%7B+frac%7B9%7D%7Bx-1%7D%26gt%3B0+%7D%7D+right.+%3D%26gt%3Bx-1%26gt%3B0%3B+x%26gt%3B1%3B+xin%281%3B%2Binfty%29%3B%5C%0Alog_%7B3%7D%5E2%28x-1%29%2Blog_%7B3%7D%289%5E2%29-log_%7B3%7D%28%28x-1%29%5E2%29%3D7%3B%5C%0Alog_%7B3%7D%5E2%28x-1%29-2log_%7B3%7D%28x-1%29%3D7-4%3D3%3B%5C%0Alog_%7B3%7D%28x-1%29%3Dt%3B%5C%0At%5E2-2t-3%3D0%3B%0AD%3D4%2B12%3D16%3B%5C%0At_%7B1%7D%3D+frac%7B2-4%7D%7B2%7D%3D-1%3D%26gt%3Blog_%7B3%7D%28x-1%29%3Dlog_%7B3%7D%283%5E%7B-1%7D%29%3D%26gt%3B+x-1%3D+frac%7B1%7D%7B3%7D%3D%26gt%3Bx%3D-+frac%7B2%7D%7B3%7D%5Cnotin%281%3B%2Binfty%29%3B%5C%0At_%7B2%7D%3D+frac%7B2%2B4%7D%7B2%7D%3D3%3B%3D%26gt%3Blog_%7B3%7D%28x-1%29%3Dlog_%7B3%7D%283%5E3%29%3B%3D%26gt%3Bx-1%3D27%3D%26gt%3B%5C%0Axin%280%3B%2Binfty%29%3B%5C%0A+%0A++++ "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;\](https://tex.z-dn.net/?f=x%5E%7Bln%28x%29%7D%3De%5E2cdot+x%3B%5C%0Ax%26gt%3B0%3B%5C+%0Ae%5E%7Bln%28x%5E%7Bln%28x%29%7D%29%7D%3De%5E2x%3B%5C%0Ae%5E%7Bln%5E2%28x%29%7D%3De%5E2x%3B%5C%0Aln%28e%5E%7Bln%5E2%28x%29%7D%29%3Dln%28e%5E2%29%2Bln%28x%29%3B%5C%0Aln%5E2%28x%29-ln%28x%29-2%3D0%3B%5C%0AD%3D1%2B8%3D9%3B%5C%0A+frac%7B1-3%7D%7B2%7D%3D-1%3B+x%3De%5E%7B%28-1%29%7Din%280%3B%2Binfty%29%5C%0A++frac%7B1%2B3%7D%7B2%7D%3D2%3B+x%3De%5E2in%280%3B%2Binfty%29%5C%0Ax%3De%5E%7B-1%7D%3B+x%3De%5E2%3B%5C%0A+%0A%0A "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)](https://tex.z-dn.net/?f=log_%7B+frac%7B1%7D%7B3%7D%7D%28x-2%29%26gt%3B-3log_%7B+frac%7B1%7D%7B5%7D%7D+%28++sqrt%5B3%5D%7B+frac%7B1%7D%7B5%7D%7D++%29%3B%5C%0Ax%26gt%3B2%3B+xin%282%3B%2Binfty%29%3B%5C%0Alog_%7B+frac%7B1%7D%7B3%7D%7D%28x-2%29%26gt%3B-1%3B%5C%0Alog_%7B+frac%7B1%7D%7B3%7D%7D%28x-2%29%26gt%3Blog_%7B+frac%7B1%7D%7B3%7D%7D%28%28frac%7B1%7D%7B3%7D%29%5E-1%29+%3B%5C%0Ax-2%26lt%3B3%3B%0Ax%26lt%3B5%3B%0A%5C+%C2%A0xin%282%3B5%29 "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<>
2.a)
b)
x=28;
c)
3.
a)
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<>
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