Question

помогите решить?!!

1) lg^2 x -lg ^2 (10x)=

 2) 6-lg^2 (100x)=

Answer 1

$lg^2 x-lg^2 {10 x}=(lg x-lg (10x))(lg x+lg(10x))=lg frac{x}{10x} lg (x*10x)=lg frac{1}{10} *lg (10x^2)=lg 10^{-1} *(lg (10)+lg (x^2))=-1*(1+2lg x)=-1-2lg x$

 

$6-lg^2 {100x}=6-(lg 100+lg x)^2=6-(2+lg x)^2=6-4-4lg x-lg^2 x=-lg^2 x-4lg x+2$

 

$lg^2 x-lg^2 (10x)=6-lg^2 (100x)=0;\ x>0; 10x>0; 100x>0;\ x>0;\ -1-2lg x=-lg^2 -4lg x+2;\ lg^2 x+4lg x-2lg x-1-2=0;\ lg^2 x+2lg x-3=0;\ (lg x+3)*(lg x-1)=0;\\ lg x+3=0;\ lg x=-3;\ x_1=10^{-3}=0.001;\\ lg x-1=0;\ lg x=1; x_2=10^1=10; x_1*x_2=0.001*10=0.01$