Question

Tổng các nghiệm của phương trình:

\(log_2\left(2^x-1\right).log_4\left(2^{x+1}-2\right)=1\)

Answer 1

ĐKXĐ: \(x>0\)

\(log_2\left(2^x-1\right).\left[log_4\left[2\left(2^x-1\right)\right]\right]=1\)

\(\Leftrightarrow log_2\left(2^x-1\right).\left[\dfrac{1}{2}log_22+\dfrac{1}{2}log_2\left(2^x-1\right)\right]=1\)

Đặt \(log_2\left(2^x-1\right)=t\)

\(\Rightarrow t\left(\dfrac{1}{2}+\dfrac{1}{2}t\right)=1\)

\(\Leftrightarrow t^2+t-2=0\Rightarrow\left[{}\begin{matrix}t=1\\t=-2\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}log_2\left(2^x-1\right)=1\\log_2\left(2^x-1\right)=-2\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}2^x=3\\2^x=\dfrac{5}{4}\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=log_23\\x=log_2\dfrac{5}{4}\end{matrix}\right.\) \(\Rightarrow log_23+log_2\dfrac{5}{4}=log_2\dfrac{15}{4}\)