Równania i układy równań logarytmicznych

[Einveru]

a) 2\(\log_{x}3\) \(\cdot\) \(log_{3x}3\) = \(log_{9√x}3\)


b) Układ

\(log_{2}(1 + \frac{x}{y}) = 2 - log_{2}y\)
\(log_{4}(xy)\) = 1

[eresh]

a) 2\(\log_{x}3\) \(\cdot\) \(log_{3x}3\) = \(log_{9√x}3\)

\(D=(0,\infty)\setminus\{1,\frac{1}{3},\frac{1}{81}\}\)

\(2\log_x3\log_{3x}3=\log_{9\sqrt{x}}3\\
\frac{2}{\log_3x}\cdot\frac{1}{\log_3(3x)}=\frac{1}{\log_3(9\sqrt{x})}\\
2\log_39\sqrt{x}=\log_3x(\log_3x+1)\\
2\log_3\sqrt{x}+4=\log_3^2x+\log_3x\\
\log_3^2x=4\\
\log_3x=2\;\; \vee \;\;\;\log_3x=-2\\
x=9\;\;\;\vee\;\;x=\frac{1}{9}\)

[eresh]

b) Układ

\(log_{2}(1 + \frac{x}{y}) = 2 - log_{2}y\)
\(log_{4}(xy)\) = 1

\(1+\frac{x}{y}>0\;\;\; \wedge \;\;\;y>0\;\;\; \wedge xy>0\\
\frac{x}{y}>-1\;\;\; \wedge \;\;\;y>0\;\;\; \wedge x>0\\
x>0\;\;\wedge\;\;y>0\)

\(\log_4(xy)=1\\
xy=4\\
y=\frac{4}{x}\)


\(\log_2(1+\frac{x}{y})=2-\log_2y\\
\log_2(1+\frac{x}{\frac{4}{x}})=2-\log_2\frac{4}{x}\\
\log_2(1+\frac{x^2}{4})=\log_24-\log_2\frac{4}{x}\\
\log_2(1+\frac{x^2}{4})=\log_2x\\
1+\frac{x^2}{4}=x\\
x^2-4x+4=0\\
x=2\\
y=\frac{4}{2}=2\)