Ответ: [tex]\frac{x+y}{x-y}=\frac{5}{3}[/tex]
Пошаговое решение:
[tex]\left \{ {{3^{x}=4} \atop {9^{y}=2}} \right \\\left \{ {{log_{3}3^{x}=log_{3}4} \atop {log_{3}3^{2y}=log_{3}2}} \right \\\left \{ {{x=log_{3}4} \atop {2y=log_{3}2}} \right\\\left \{ {{x=log_{3}4} \atop {y=\frac{log_{3}2}{2}}} \right[/tex]
[tex]\frac{x+y}{x-y}=\frac{log_{3}4+\frac{log_{3}2}{2}}{log_{3}4-\frac{log_{3}2}{2}}=\frac{\frac{2log_{3}4+log_{3}2}{2}}{\frac{2log_{3}4-log_{3}2}{2}}=\frac{\frac{log_{3}4^{2}+log_{3}2}{2}}{\frac{log_{3}4^{2}-log_{3}2}{2}}=\frac{\frac{log_{3}16+log_{3}2}{2}}{\frac{log_{3}16-log_{3}2}{2}}=\frac{log_{3}16*2}{log_{3}16:2}=\frac{log_{3}2^{5}}{log_{3}2^{3}}=\frac{5log_{3}2}{3log_{3}2}=\frac{5}{3}[/tex]
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Ответ: [tex]\frac{x+y}{x-y}=\frac{5}{3}[/tex]
Пошаговое решение:
[tex]\left \{ {{3^{x}=4} \atop {9^{y}=2}} \right \\\left \{ {{log_{3}3^{x}=log_{3}4} \atop {log_{3}3^{2y}=log_{3}2}} \right \\\left \{ {{x=log_{3}4} \atop {2y=log_{3}2}} \right\\\left \{ {{x=log_{3}4} \atop {y=\frac{log_{3}2}{2}}} \right[/tex]
[tex]\frac{x+y}{x-y}=\frac{log_{3}4+\frac{log_{3}2}{2}}{log_{3}4-\frac{log_{3}2}{2}}=\frac{\frac{2log_{3}4+log_{3}2}{2}}{\frac{2log_{3}4-log_{3}2}{2}}=\frac{\frac{log_{3}4^{2}+log_{3}2}{2}}{\frac{log_{3}4^{2}-log_{3}2}{2}}=\frac{\frac{log_{3}16+log_{3}2}{2}}{\frac{log_{3}16-log_{3}2}{2}}=\frac{log_{3}16*2}{log_{3}16:2}=\frac{log_{3}2^{5}}{log_{3}2^{3}}=\frac{5log_{3}2}{3log_{3}2}=\frac{5}{3}[/tex]