Question

Помогите решить логарифмические уравнения

Answer 1

$8)\; \; lg5+lg(x+10)=1-lg(2x-1)+lg(21x-20)\\\\ODZ:\; \; \left \{ {{x+10>0\; ,\; 2x-1>0} \atop {21x-20>0}} \right.\; \; \left \{ {{x>-10\; ,\; x>0,5} \atop {x>\frac{20}{21}}} \right.\; \; \to \; \; x>\frac{20}{21}\\\\lg5+lg(x+10)+lg(2x-1)=lg10+lg(21x-20)\\\\lg\Big (5(x+10)(2x-1)\Big )=lg\Big (10\cdot (21x-20)\Big )\\\\5\cdot (2x^2+19x-10)=10\cdot (21x-20)\\\\2x^2+19x-10=2(21x-20)\\\\2x^2-23x+30=0\; \; ,\; \; D=289=17^2\; \; ,\; \; x_{1,2}=\frac{23\pm 17}{4}\\\\x_1=\frac{3}{2}\; ,\; \; x_2=10\\\\Otvet:\; \; x_1=\frac{3}{2}\; ,\; \; x_2=10$

$9)\; \; log_2182-2log_2\sqrt{5-x}=log_2(11-x)+1\\\\ODZ:\; \; \left \{ {{5-x>0} \atop {11-x>0}} \right.\; \; \left \{ {{x<5} \atop {x<11}} \right.\; \; \to \; \; \underline {\; x<5\; }\\\\log_2182=log(11-x)+log_22+log_2(5-x)\\\\182=2(11-x)(5-x)\\\\91=55-16x+x^2\\\\x^2-16x-36=0\; \; ,\; \; D/4=8^2+36=100\; ,\; \; x_{1,2}=8\pm 10\\\\x_1=-2\in ODZ\; \; ,\; x_2=18\notin ODZ\\\\Otvet:\; \; x=-2\; .$

$10)\; \; log_5\sqrt{x-9}-log_510+log_5\sqrt{2x-1}=0\\\\ODZ:\; \; \left \{ {{x-9>0} \atop {2x-1>0}} \right.\; \; \left \{ {{x>9} \atop {x>0,5}} \right.\; \; \to \; \; \underline {\; x>9\; }\\\\log_5\sqrt{x-9}+log_5\sqrt{2x-1}=log_510\\\\\sqrt{x-9}\cdot \sqrt{2x-1}=10\\\\(x-9)(2x-1)=100\\\\2x^2-19x+9=100\\\\2x^2-19x-91=0\; \; ,\; \; D=725\; ,\; \; x_{1,2}=\frac{19\pm \sqrt{725}}{4}=\frac{19\pm 5\sqrt{29}}{4}\\\\x_1=\frac{19-5\sqrt{29}}{4}\approx -1,98\notin ODZ\; \; ,\; \; x_2=\frac{19+5\sqrt{29}}{4}\approx 11,48\in ODZ\\\\Otvet:\; \; x=\frac{19+5\sqrt{29}}{4}$