Логаритмични уравнения

[Tinna]

1 зад. [tex]\frac{1}{3 }log_{2} (3x-4)^{6}.log_{2}x^{3}=8log_{2} ^{2}\sqrt[]{x} +log_{2} ^{2} (3x-4)^{2}[/tex]
2 зад. [tex](x-4)^{2}log_{4}(x-1)-2log_{4}(x-1)^{2}=(x-4)^{2}log_{x-1} 4 -2log_{x-1}16[/tex]

[Реклама]

[_sssss]

[tex]\normal log_{2}(3x-4)^2 = u \\ log_{2}x=v[/tex]

[tex]\normal 3uv=2v^2 + u^2 \\ 2\( \frac{v}{u} \)^2 -3\frac{v}{u} + 1=0[/tex]

[tex]\normal \frac{v}{u}=1 \Leftrightarrow 9x^2 - 25x +16=0 \Leftrightarrow x=1, \; x=\frac{16}{9}\\ \frac{v}{u}=\frac{1}{2} \Leftrightarrow x=|3x-4|\Leftrightarrow x=1, \; x=2[/tex]

[tex]\normal x=\frac{16}{9}, \; x=2[/tex]

[Spider Iovkov]

[tex](x-4)^2\log_{4}(x-1)-2\log_{4}(x-1)^2=(x-4)^2\log_{x-1}4-2\log_{x-1}16[/tex]
[tex]{\cyr D.M.} x>1, x\neq 2[/tex]
[tex](x-4)^2\log_{4}(x-1)-4\log_{4}(x-1)=(x-4)^2.\frac{1}{\log_{4}(x-1)}-\frac{2.2}{\log_{4}(x-1)} \Leftrightarrow[/tex]
[tex]\Leftrightarrow (x-4)^2\log_{4}(x-1)-(x-4)^2.\frac{1}{\log_{4}(x-1)}=4\log_{4}(x-1)-\frac{4}{\log_{4}(x-1)} \Leftrightarrow[/tex]
[tex]\Leftrightarrow (x-4)^2[\log_{4}(x-1)-\frac{1}{\log_{4}(x-1)}]=4[\log_{4}(x-1)-\frac{1}{\log_{4}(x-1)}][/tex]
І случай. [tex]\log_{4}(x-1)-\frac{1}{\log_{4}(x-1)}=0 \Rightarrow \log_{4}^2(x-1)=1 \Leftrightarrow \log_{4}(x-1)=\pm 1 \Leftrightarrow x_{1}=5, x_{2}=\frac{5}{4}[/tex]
ІІ случай. [tex]\log_{4}(x-1)-\frac{1}{\log_{4}(x-1)} \neq 0 \Rightarrow (x-4)^2=4 \Leftrightarrow (x-4)^2-2^2=0 \Leftrightarrow (x-2)(x-6)=0 \Leftrightarrow x_{3}=2, x_{4}=6[/tex]
[tex]x_{1}, x_{2}, x_{4}\in {\cyr D.M.}, x_{3}\cancel \in {\cyr D.M.}[/tex]