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Olgalupascu14id

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log2x=log4(3x−2)                C.E. {3x−2 \textgreater 0x \textgreater 0⇒{x \textgreater 32x \textgreater 0log2x=log42log4xlog42log4x=log4(3x−2)                   log42=log44=21log44=2121log4x=log4(3x−2)2log4x=log4(3x−2)log4x2=log4(3x−2)x2=3x−2x2−3x+2=0a=1, b=−3, c=2

\begin{gathered}\displaystyle \Delta=b^2-4ac=(-3)^2-4 \cdot 1 \cdot 2=9-8=1\ \textgreater \ 0 \\ \\ x_1= \frac{-b+ \sqrt{\Delta} }{2a} =\frac{3+ \sqrt{1} }{2 \cdot 1} = \frac{3+1}{2} = \frac{4}{2} =\boxed2 \\ \\ x_2=\frac{-b- \sqrt{\Delta} }{2a} = \frac{3- \sqrt{1} }{2 \cdot 1} = \frac{3-1}{2} = \frac{2}{2} =\boxed1\end{gathered}Δ=b2−4ac=(−3)2−4⋅1⋅2=9−8=1 \textgreater 0x1=2a−b+Δ=2⋅13+1=23+1=24=2x2=2a−b−Δ=2⋅13−1=23−1=22=1

PutereDinuid

[tex]\log_2(x)=\log_4(3x-2) , \ x \in(\frac{2}{3}, \ +\infty) \\ \log_2(x)=\log_{2^2}(3x-2) \\ \log_2(x)=\frac{1}{2} \log_2(3x-2) \\ \log_2(x)=\log_2((3x-2)^{\frac{1}{2}}) \\ \Rightarrow x=(3x-2)^{\frac{1}{2} }\ \Longleftrightarrow \ x^2=((3x-2)^{\frac{1}{2}})^2 \\ \Rightarrow x^2=3x-2 \ \Longleftrightarrow \ x2-3x+2=0 \\ x^2-x-2x+2=0 \\ x(x-1)-2(x-1)=0 \\ (x-1)(x-2)=0 \\ \Rightarrow \ \left \{ {{x=1} \atop {x=2}} \right , \ x \in (\frac{2}{3}, \ +\infty) \\ \Rightarrow \ \boxed{x_1=1 \ ,\ x_2=2}}[/tex]

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