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Question-172349




Question Number 172349 by Mikenice last updated on 25/Jun/22
Answered by mr W last updated on 26/Jun/22
log 3 ×(log 3+log x)=log 4 ×(log 4+log y)  log 4 ×log x=log 3×log y  let X=log x, Y=log y  a=log 3, b=log 4  ⇒a(a+X)=b(b+Y)  ⇒bX=aY  ⇒X=−a ⇒log x=−log 3 ⇒x=(1/3)  ⇒Y=−b ⇒log y=−log 4 ⇒y=(1/4)
$$\mathrm{log}\:\mathrm{3}\:×\left(\mathrm{log}\:\mathrm{3}+\mathrm{log}\:{x}\right)=\mathrm{log}\:\mathrm{4}\:×\left(\mathrm{log}\:\mathrm{4}+\mathrm{log}\:{y}\right) \\ $$$$\mathrm{log}\:\mathrm{4}\:×\mathrm{log}\:{x}=\mathrm{log}\:\mathrm{3}×\mathrm{log}\:{y} \\ $$$${let}\:{X}=\mathrm{log}\:{x},\:{Y}=\mathrm{log}\:{y} \\ $$$${a}=\mathrm{log}\:\mathrm{3},\:{b}=\mathrm{log}\:\mathrm{4} \\ $$$$\Rightarrow{a}\left({a}+{X}\right)={b}\left({b}+{Y}\right) \\ $$$$\Rightarrow{bX}={aY} \\ $$$$\Rightarrow{X}=−{a}\:\Rightarrow\mathrm{log}\:{x}=−\mathrm{log}\:\mathrm{3}\:\Rightarrow{x}=\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\Rightarrow{Y}=−{b}\:\Rightarrow\mathrm{log}\:{y}=−\mathrm{log}\:\mathrm{4}\:\Rightarrow{y}=\frac{\mathrm{1}}{\mathrm{4}} \\ $$

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