Question

va rog mult.ex 1 si 2
De la ex 1 punctele d,f,g,h,i,j
Am nevoie urgent!
multumesc si dau coroana !

Answer 1

$1)\:d)\:\lg\frac{2}{x-1}=\lg x\\C.E. \left \{ {{\frac{2}{x-1}\ \textgreater \ 0,\:x-1 \neq 0} \atop {x\ \textgreater \ 0}} \right. \\\frac{2}{x-1}=x\leftrightarrow x(x-1)=2\leftrightarrow x^2-x-2=0\\\Delta=1+8=9\\x_{1,2}=\frac{1\pm3}{2}\\x_1=-1\ \textless \ 0\\x_2=2\text{ verifica C.E.}$

[tex]f)\:\lg(x^2-4x)=\lg3(2-x)\\ C.E. \left \{ {{x^2-4x\ \textgreater \ 0} \atop {2-x\ \textgreater \ 0}} \right. \\ x^2-4x=3(2-x)\leftrightarrow x^2-4x=6-3x\leftrightarrow x^2-x-6=0\\\Delta=1+24=25\\x_{1,2}=\frac{1\pm5}{2}\\x_1=-2\text{ verifica C.E.}\\x_2=3\text{ nu verifica C.E.}[/tex]

$g)\:2\lg(2x+1)=\lg(x^2+7x+61)\leftrightarrow\lg(2x+1)^2=\lg(x^2+7x+61)\\C.E. \left \{ {{2x+1\ \textgreater \ 0} \atop {x^2+7x+61\ \textgreater \ 0}} \right. \\(2x+1)^2=x^2+7x+61\\4x^2+4x+1=x^2+7x+61\\3x^2-3x-60=0\\x^2-x-20=0\\\Delta=1+80=81\\x_{1,2}=\frac{1\pm9}{2}\\x_1=-4\text{ nu verifica C.E.}\\x_2=5\text{ verifica C.E.}$

$h)\:\log_2(3x+5)=\log_28(x+1)\\C.E. \left \{ {{3x+5\ \textgreater \ 0} \atop {x+1\ \textgreater \ 0}} \right. \\3x+5=8(x+1)\\3x+5=8x+8\\-5x=3\leftrightarrow x=-\frac{3}{5}\text{ verifica C.E.}$

[tex]i)\:\log_3(4^x+3\cdot2^x)=\log_3(2^{x+2}+2)\\4^x+3\cdot2^x=2^{x+2}+2\\2^{2x}+3\cdot2^x-2^2\cdot2^x-2=0\\ 2^{2x}-2^x-2=0\\\text{Subtitutie: }2^x=t,\:t\ \textgreater \ 0\\\text{Ecuatia devine: }t^2-t-2=0\\\Delta=1+8=9\\t_{1,2}=\frac{1\pm3}{2}\\t_1=-1\ \textless \ 0\\t_2=2\\2^x=2\leftrightarrow x=1[/tex]

$j)\:\lg(10^{2x+1}+7\cdot10^{x+1})=\lg(10^{x+2}-20)\\10^{2x+1}+7\cdot10^{x+1}=10^{x+2}-20\\10\cdot10^{2x}+7\cdot10\cdot10^x-10^2\cdot10^x+20=0|:10\\10^{2x}-3\cdot10^x+2=0\\\text{Substitu\c tie: }10^x=t,\:t\ \textgreater \ 0\\\text{Ecua\c tia devine: }t^2-3t+2=0\\\Delta=9-8=1\\t_{1,2}=\frac{3\pm1}{2}\\t_1=1\\t_2=2\\10^x=1\leftrightarrow x=0\\10^x=2\leftrightarrow x=\lg2\\x\in\{0,\lg2\}$

$2)\:a)\:\log_4x=2\leftrightarrowx=4^2=16\text{ verifica C.E.}\\C.E.\:x\ \textgreater \ 0$

$b)\:\lg(x-1)=-1\leftrightarrow x-1=10^{-1}\leftrightarrow x=\frac{1}{10}+1=\frac{11}{10}\text{ verifica C.E.}\\C.E.\:x-1\ \textgreater \ 0$