Question

giải pt: log_\(\sqrt{2}\)(\(^{4^x-3.2^{x+1}+2}\))=2x+4

Answer 1

\(\Rightarrow4^x-3.2^{x+1}+2=\sqrt{2}^{2\left(x+2\right)}\)

\(\Leftrightarrow4^x-6.2^x+2=2^{x+2}=4.2^x\)

Đặt \(2^x=a>0\Rightarrow a^2-6a+2=4a\)

\(\Leftrightarrow a^2-10a+2=0\Rightarrow\left[{}\begin{matrix}a=5+\sqrt{23}\\a=5-\sqrt{23}\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}2^x=5+\sqrt{23}\\2^x=5-\sqrt{23}\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=log_2\left(5+\sqrt{23}\right)\\x=log_2\left(5-\sqrt{23}\right)\end{matrix}\right.\)