33.
ĐKXĐ: \(x>3+\sqrt{2}\)
\(\log\left(x^2-6x+7\right)=\log\left(x-3\right)\)
\(\Rightarrow x^2-6x+7=x-3\)
\(\Leftrightarrow x^2-7x+10=0\)
\(\Rightarrow\left[{}\begin{matrix}x=2\left(loại\right)\\x=5\end{matrix}\right.\)
34.
ĐKXĐ: \(x>\dfrac{3}{2}\)
\(\log_{3}(x^2+4x)+\log_{\frac{1}{3}}(2x-3)=0\)
\(\Leftrightarrow \log_{3}(x^2+4x)+\log_{3^{-1}}(2x-3)=0\)
\(\Leftrightarrow\log_3\left(x^2+4x\right)-\log_3\left(2x-3\right)=0\)
\(\Leftrightarrow\log_3\left(x^2+4x\right)=\log_3\left(2x-3\right)\)
\(\Rightarrow x^2+4x=2x-3\)
\(\Rightarrow x^2+2x+3=0\)
Phương trình vô nghiệm
35.
ĐKXĐ: \(x>-1\)
\(\ln\left(x+1\right)+\ln\left(x+3\right)=\ln\left(x+7\right)\)
\(\Leftrightarrow\ln\left(x+1\right)\left(x+3\right)=\ln\left(x+7\right)\)
\(\Rightarrow\left(x+1\right)\left(x+3\right)=x+7\)
\(\Leftrightarrow x^2+3x-4=0\Rightarrow\left[{}\begin{matrix}x=1\\x=-4\left(loại\right)\end{matrix}\right.\)
36.
ĐKXĐ: \(x>0\)
\(\log_2x+\log_{2^2}x+\log_{2^3}x=11\)
\(\Leftrightarrow\log_2x+\dfrac{1}{2}\log_2x+\dfrac{1}{3}\log_2x=11\)
\(\Leftrightarrow\dfrac{11}{6}\log_2x=11\)
\(\Leftrightarrow\log_2x=6\)
\(\Rightarrow x=2^6=64\)
37.
ĐXĐK \(x>0;x\ne1\)
\(\log_2x+3\log_x2=4\)
\(\Leftrightarrow\log_2x+\dfrac{3}{\log_2x}=4\)
\(\Leftrightarrow\log_2^2x+3=4\log_2x\)
\(\Leftrightarrow\log_2^2x-4\log_2x+3=0\)
\(\Rightarrow\left[{}\begin{matrix}\log_2x=1\\\log_2x=3\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=2\\x=8\end{matrix}\right.\)
38.
ĐKXĐ: \(x>0;x\ne\left\{10^4;\dfrac{1}{100}\right\}\)
\(\dfrac{1}{4-\log x}+\dfrac{2}{2+\log x}=1\)
\(\Rightarrow2+\log x+2\left(4-\log x\right)=\left(4-\log x\right)\left(2+\log x\right)\)
\(\Leftrightarrow10-\log x=8+2\log x-\log^2x\)
\(\Leftrightarrow\log^2x-3\log x+2=0\)
\(\Rightarrow\left[{}\begin{matrix}\log x=1\\\log x=2\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=10\\x=100\end{matrix}\right.\)
39.
ĐKXĐ: \(x>0\)
\(\log_3\left(x+2\right)^2+2\log_3x=2\)
\(\Rightarrow2\log_3\left(x+2\right)+2\log_3x=2\)
\(\Leftrightarrow\log_3\left(x+2\right)+\log_3x=1\)
\(\Leftrightarrow\log_3x\left(x+2\right)=1\)
\(\Rightarrow x\left(x+2\right)=3\)
\(\Leftrightarrow x^2+2x-3=0\Rightarrow\left[{}\begin{matrix}x=1\\x=-3\left(loại\right)\end{matrix}\right.\)
40.
ĐKXĐ: \(x>\log_3\left(\dfrac{1}{2}\right)\)
\(\log_4\left(3.2^x-1\right)=x-1\)
\(\Leftrightarrow3.2^x-1=4^{x-1}\)
\(\Leftrightarrow3.2^x-1=2^{2x-2}\)
\(\Leftrightarrow12.2^x-4=2^{2x}\)
\(\Leftrightarrow\left(2^x\right)^2-12.2^x+4=0\)
\(\Rightarrow\left[{}\begin{matrix}2^x=6+4\sqrt{2}\\2^x=6-4\sqrt{2}\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=\log_2\left(6+4\sqrt{2}\right)\\x=\log_2\left(6-4\sqrt{2}\right)\end{matrix}\right.\)
\(\Rightarrow\log_2\left(6+4\sqrt{2}\right)+\log_2\left(6-4\sqrt{2}\right)=\log_24=2\)
41.
ĐKXĐ: \(x>0\)
Đặt \(\log_2x=t\)
\(\Rightarrow t^2+t+m=0\)
Pt có nghiệm khi:
\(\Delta=1-4m\ge0\Rightarrow m\le\dfrac{1}{4}\)
42.
ĐKXĐ: \(x>-\dfrac{3}{2}\)
\(\log_3\left(2x+3\right)=3\)
\(\Leftrightarrow2x+3=27\)
\(\Leftrightarrow x=12\)
43.
ĐKXĐ: \(0< x< 6\sqrt{2}\)
\(\log\left(72-x^2\right)=2\log x\)
\(\Rightarrow\log\left(72-x^2\right)=\log x^2\)
\(\Rightarrow72-x^2=x^2\)
\(\Rightarrow x^2=36\)
\(\Rightarrow\left[{}\begin{matrix}x=6\\x=-6\left(loại\right)\end{matrix}\right.\)
