Równanie 1 [latex]log_{x}cfrac{x+2}{x}=1[/latex] Dziedzina: [latex]{x>0}qquadlandqquad{x eq1}qquadlandqquad{cfrac{x+2}{x}>0}[/latex] [latex]x>0Rightarrow{x+2>0}[/latex] [latex]xin(0,1)cup(1,+infty)[/latex] Z definicji logarytmu: [latex]cfrac{x+2}{x}=x[/latex] [latex]x+2=x^2[/latex] [latex]x^2-x-2=0[/latex] [latex]Delta=(-1)^2-4cdot1cdot(-2)=1+8=9[/latex] [latex]x_1=cfrac{1-3}{2}=-1 otin D[/latex] [latex]x_2=cfrac{1+3}{2}=2[/latex] Równanie 2: [latex]log_{x^2} (x+2) = 1[/latex] Dziedzina: [latex]x^2>0qquadlandqquad{x^2 eq1}qquadlandqquad{x+2>0}[/latex] [latex]x eq0qquadlandqquad{x eq1}qquadlandqquad{x eq-1}qquadlandqquad{x>-2}[/latex] [latex]xin(-2,-1)cup(-1,0)cup(0,1)cup(1,+infty)[/latex] Z definicji logarytmu: [latex]x+2=x^2[/latex] [latex]x^2-x-2=0[/latex] [latex]Delta=(-1)^2-4cdot1cdot(-2)=1+8=9[/latex] [latex]x_1=cfrac{1-3}{2}=-1 otin D[/latex] [latex]x_2=cfrac{1+3}{2}=2[/latex] Równanie 3: [latex]log_2(log_4 x) = 1[/latex] Dziedzina: [latex]x>0qquadlandqquad{log_4 x > 0}[/latex] [latex]x>0qquadlandqquad{x>1}[/latex] [latex]xin(1,+infty)[/latex] Z definicji logarytmu: [latex]log_4 x=2[/latex] [latex]x=4^2=16[/latex] Równanie 4: [latex]log_4 (x-3) - 2 = log_4 (x-1) - log_4 8[/latex] Dziedzina: [latex]x-3>0qquadlandqquad{x-1>0}[/latex] [latex]xin(3,+infty)[/latex] [latex]log_4(x-3)-log_4 16=log_4 cfrac{x-1}{8}[/latex] [latex]log_4 cfrac{x-3}{16}=log_4 cfrac{x-1}{8}[/latex] [latex]cfrac{x-3}{16}=cfrac{x-1}{8}[/latex] [latex]8(x-3)=16(x-1)[/latex] [latex]8x-24=16x-16[/latex] [latex]-8x=8[/latex] [latex]x=-1 otin D[/latex] Równanie 5: [latex]log 5 + log(x+10) = 1 - log (2x - 1) + log (21x - 20)[/latex] Dziedzina: [latex]x+10>0qquadlandqquad2x-1>0qquadlandqquad21x-20>0[/latex] [latex]x>-10qquadlandqquad{x>0,5}qquadlandqquad{x>cfrac{20}{21}}[/latex] [latex]xinleft(cfrac{20}{21},+infty ight)[/latex] [latex]log 5(x+10)=log cfrac{10(21x-20)}{2x-1}[/latex] [latex]5x+50=cfrac{210x-200}{2x-1}[/latex] [latex](5x+50)(2x-1)=210x-200[/latex] [latex]10x^2-5x+100x-50-210x+200=0[/latex] [latex]10x^2-115x+150=0[/latex] [latex]Delta=(-115)^2-4cdot10cdot150=13225-6000=7225[/latex] [latex]sqrt{Delta}=85[/latex] [latex]x_1=cfrac{115-85}{20}=cfrac{3}{2}[/latex] [latex]x_2=cfrac{115+85}{20}=10[/latex]
