1.
a.
ĐKXĐ: \(x^2-1>0\Rightarrow\left[{}\begin{matrix}x>1\\x< -1\end{matrix}\right.\)
\(log_2\left(x^2-1\right)=3\)
\(\Rightarrow x^2-1=8\)
\(\Leftrightarrow x^2=9\)
\(\Rightarrow x=\pm3\) (tm)
b.
ĐKXĐ: \(x>0\)
\(log_3x+log_{\sqrt{3}}x+log_{\dfrac{1}{3}}x=6\)
\(\Leftrightarrow log_3x+2log_3x-log_3x=6\)
\(\Leftrightarrow log_3x=3\)
\(\Rightarrow x=3^3=27\)
c. ĐKXĐ: \(x>0\)
\(log_{\sqrt{2}}^2x+3log_2x+log_{\dfrac{1}{2}}x=2\)
\(\Leftrightarrow\left(2log_2x\right)^2+3log_2x-log_2x=2\)
\(\Leftrightarrow4log_2^2x+2log_2x-2=0\)
\(\Rightarrow\left[{}\begin{matrix}log_2x=-1\\log_2x=\dfrac{1}{2}\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=\dfrac{1}{2}\\x=\sqrt{2}\end{matrix}\right.\)
d.
ĐKXĐ: \(x>0\)
\(log_{\dfrac{1}{2}}^24x+log_2\dfrac{x^2}{8}=8\)
\(\Leftrightarrow\left(-log_24x\right)^2+log_2x^2-log_28=8\)
\(\Leftrightarrow\left(log_2x+log_24\right)^2+2log_2x-3=8\)
\(\Leftrightarrow\left(log_2x+2\right)^2+2log_2x-11=0\)
\(\Leftrightarrow log_2^2x+6log_2x-7=0\)
\(\Rightarrow\left[{}\begin{matrix}log_2x=1\\log_2x=-7\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=2\\x=\dfrac{1}{2^7}\end{matrix}\right.\)
2.
a.
Lấy logarit cơ số 5 hai vế:
\(log_52^{x-3}+log_55^{x^2-5x+6}=0\)
\(\Leftrightarrow\left(x-3\right)log_52+x^2-5x+6=0\)
\(\Leftrightarrow\left(x-3\right)log_52+\left(x-3\right)\left(x-2\right)=0\)
\(\Leftrightarrow\left(x-3\right)\left(x-2+log_52\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=3\\x=log_52-2\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=3\\x=log_5\dfrac{2}{25}\end{matrix}\right.\)
b.
\(2^x.5^{x^2-4x}=1\)
Lấy logarit cơ số 5 hai vế:
\(log_5\left(2^x.5^{x^2-4x}\right)=log_51\)
\(\Leftrightarrow log_52^x+log_55^{x^2-4x}=0\)
\(\Leftrightarrow x.log_52+x^2-4x=0\)
\(\Leftrightarrow x.log_52+x\left(x-4\right)=0\)
\(\Leftrightarrow x\left(log_52+x-4\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x=4-log_52\end{matrix}\right.\)
