Question

помогите ❗❗❗❗
логарифмы​

Answer 1

$1. a) log_3(x-2)+log_3(x+6)=2; OD3: left { {{x-2>0} atop {x+6>0}} right. Rightarrow x > 2\\log_3((x-2)(x+6))=log_39\\(x-2)(x+6)=9,\\x^2+4x-12-9=0;\\x^2+4x-21=0.\\left { {{x_1+x_2=-4,} atop {x_1x_2=-21}} right. Rightarrow x_1=-7notin OD3, x_2=3.$

ОТВЕТ: 3.

$b) log_3^2x+log_3x^2=8; OD3: x>0\\log_3^2x+2log_3x-8=0. \\log_3x=t,\\t^2+2t-8=0Rightarrow t_1=2, t_2=-4.\\left [ {{log_3x=2} atop {log_3x=-4;}} right. left [ {{x=9} atop {x=frac{1}{81} }} right.$

ОТВЕТ: $frac{1}{81} ; 9$.

$2. log_{frac{1}{3}}(x-1) geq -2; OD3: x - 1 > 0Rightarrow x>1.\x-1leq (frac{1}{3} )^{-2},\x-1leq 9; \x leq 10$

ОТВЕТ: (1; 10].

$3. frac{1}{5-lg x}+frac{2}{1+lg x}=1.\ OD3: left { {{lg xneq 5} atop {lg xneq -1}} right. Rightarrow xneq 0,1; 100000.\\lg x = t neq -1;5.\\frac{1}{5-t}+frac{2}{1+t}=1,\\ frac{1+t+10-2t}{(5-t)(1+t)}=1,\\ -t+11=5+4t-t^2,\\t^2-5t+6=0Rightarrow x_1=2,x_2=3.\left [ {{lg x =2} atop {lg x=3}} right. Rightarrow left [ {{x=100} atop {x=1000}} right.$

ОТВЕТ: 100; 1000.

$4. log_3x-log_x3leq 1,5. OD3: left { {{x>0} atop {xneq 1}} right. \\log_3x-frac{1}{log_3x}-1,5leq 0. \\ log_3x=tneq 1.\\t-frac{1}{t}-1,5leq 0|cdot2t\\ 2t^2-3t-1leq 0;\\D=(-3)^2-4cdot4cdot(-1)=9+16=25,\\t_{1,2}=frac{3pm5}{4}; t_1=2, t_2=-0,5.\\ tin[-0,5;2].\\ -0,5leq log_3xleq 2\\3^{-0,5}leq xleq 3^2\\frac{sqrt3}{3} leq xleq 9$

С учетом условия x > 0, x ≠ 1, x ∈ (0; √3/3] ∪ (1; 3].

ОТВЕТ: (0; √3/3] ∪ (1; 3]