Question

1. Вычислите: $2^{ frac{1}{ log_{5} 2} } + log_{3}log _{3} sqrt[3]{ sqrt[3]{3} }$
2. Решите уравнение: $3* 16^{x}+2* 81^{x}=5* 36^{x}$
3. Решите неравенство: $log_{3}( x^{2} -2x-2) leq 0$
4. Решите систему уравнений: $left { {{ 3^{x}* 2^{y} =972 } atop { log_{ sqrt{3} } (x-y)=2}} right.$

Answer 1

$1) 2^{frac{1}{log_{5} 2} } + log_{3}log _{3} sqrt[3]{sqrt[3]{3}} =2^{log_{2} 5} + log_{3}log _{3} 3^{ frac{1}{9}} = 5 + log_{3}( frac{1}{9}log _{3} 3) =\= 5 + log_{3}3^{-2} = 5 -2log_{3}3 = 5-2=3;$

$3cdot16^{x}+2cdot81^{x}=5cdot36^{x}, \ 3cdot2^{4x}+2cdot3^{4x}-5cdot(2cdot3)^{2x}=0, \ 3cdot (frac{2}{3})^{4x}-5(frac{2}{3})^{2x}+2=0, \ (frac{2}{3})^{2x}=t, \ 3t^2-5t+2=0, \ D=1, \ t_1= frac{2}{3}, x_2=1, \ (frac{2}{3})^{2x}=frac{2}{3}, \ 2x=1, \ x_1=frac{1}{2} , \ (frac{2}{3})^{2x}=1, \ 2x=0, \ x_2=0;$

$3) log_{3}( x^{2} -2x-2) leq 0, \ left { {{ x^{2} -2x-2>0,} atop { x^{2} -2x-2 leq 1;}} right. left { {{ x^{2} -2x-2>0,} atop { x^{2} -2x-3 leq 0;}} right. \ x^{2} -2x-2=0, \ D=1^2+2=3, \ x_1=1-sqrt{3},x_2=1+ sqrt{3}, \ x^{2} -2x-3=0, \ x_1=-1,x_2=3; \ left { {{(x-1+sqrt{3})(x-1-sqrt{3})>0,} atop { (x+1)(x-3) leq 0;}} right. left { {{ left [ {{x<1-sqrt{3},} atop {x>1+sqrt{3},}} right. } atop { -1 leq x leq 3;}} right.$$left [ {{-1 leq x<1-sqrt{3},} atop {1+sqrt{3}<x leq 3;}} right. \ xin[-1;1-sqrt{3})cup(1-sqrt{3};3];$

$left { {{ 3^{x}cdot 2^{y} =972, } atop { log_{ sqrt{3} } (x-y)=2;}} right. left { {{3^{x}cdot 2^{y} =972, } atop {x-y=(sqrt{3})^2;}} right. left { {{3^{y+3}cdot 2^{y} =972, } atop {x=y+3;}} right. left { {{3^3cdot3^{y}cdot 2^{y} =972, } atop {x=y+3;}} right.left { {{6^{y} =36, } atop {x=y+3;}} right. \ left { {{6^{y} =6^2, } atop {x=y+3;}} right.left { {{x=5, } atop {y=2;}} right.\ (5;2).$