Логаритмични уравнения

[Гост]

a) lg(3x+1,5) + lgx = 0
б) [tex]log_{6}[/tex] (x+3) + [tex]log_{6 }[/tex] (x+4) = 1

[ammornil]

[tex]lg(3x+1,5)+lgx=0 \\
\vspace{12} \\
DM: \hspace{12} \left|3x+1,5 >0 \\ x>0 \right, \hspace{4} \Rightarrow \hspace{4} \left|3x>-1,5 \\ x>0 \right, \hspace{4} \Rightarrow \hspace{4} \left|x>-\frac{1}{2}\\ x>0 \right, \hspace{4} \Rightarrow \hspace{4} x \in (0; +\infty) \\
\vspace{12} \\
lg(3x+1,5)+lgx=0 \hspace{4} \Rightarrow \hspace{4} lg[x(3x+1,5)]=lg1 \hspace{4} \Rightarrow \hspace{4} x(3x+1,5)=1 \hspace{4} \Rightarrow \hspace{4} \\
3x^{2}+1,5x-1=0 \hspace{2}|.2 \hspace{4} \Rightarrow \hspace{4} 6x^{2}+3x-2=0 \\
D=3^{2}-4.6.(-2)=9+48=57 \\
x_{1,2}=\frac{-3\pm \sqrt{57}}{2.6} \\
x_{1}=\frac{-1-\sqrt{57}}{6}<0 \notin DM \\
x_{2}=\frac{-3+\sqrt{57}}{6} >0 \Rightarrow \in DM[/tex]

[ammornil]

[tex]log_{6}(x+3)+log_{6}(x+4)=1 \\
\vspace{12} \\
DM: \left|x+3>0 \\ x+4>0 \right, \hspace{4} \Rightarrow \hspace{4} \left|x>-3 \\ x>-4 \right, \hspace{4} \Rightarrow \hspace{4} x \in (-3;+\infty) \\
\vspace{12} \\
log_{6}(x+3)+log_{6}(x+4)=1 \hspace{4} \Rightarrow \hspace{4} log_{6}[(x+3).(x+4)]=log_{6}6 \\
(x+3).(x+4)=6 \hspace{4} \Rightarrow \hspace{4} x^{2}+4x+3x+12=6 \hspace{4} \Rightarrow \hspace{4} x_{2}+7x+6=0 \\
D=7^{2}-4.1.6=49-24=25 \\
x_{_{1,2}}=\frac{-7\pm\sqrt{25}}{2.1}=\frac{-7\pm5}{2} \\
x_{1}=-6 \hspace{8} \notin DM \\
x_{2}=-1 \hspace{8} \in DM[/tex]

ОТГ. х=-1.