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Question Number 85826 by jagoll last updated on 25/Mar/20

$${}^{\mathrm{x}}\log \left(\mathrm{xy}\right).{}^{\mathrm{y}}\log \left(\mathrm{xy}\right)+{}^{\mathrm{x}}\log (\mathrm{x}-\mathrm{y}){.}^{\mathrm{y}}\log (\mathrm{x}-\mathrm{y})=0$$$$\mathrm{find}\mathrm{x}+\mathrm{y}$$

Commented by john santu last updated on 25/Mar/20

$${}^x\log \left(xy\right).{}^y\log \left(xy\right)=-{}^x\log (x-y){.}^y\log (x-y)$$$${}^{(x-y)}\log \left(xy\right)=-{}^{\left(xy\right)}\log (x-y)$$$${}^{(x-y)}\log \left(xy\right)={}^{\frac{1}{xy}}\log (x-y)$$$$\Rightarrow \{\begin{array}{l}x-y=\frac{1}{xy}\\ xy=x-y\end{array}$$$$\Rightarrow xy=1\wedge x-y=1$$$$y=\frac{1}{x}\Rightarrow x-\frac{1}{x}=1$$$$x^2-x-1=0\Rightarrow x=\frac{1+\sqrt{5}}{2}$$$$\wedge y=\frac{2}{\sqrt{5}+1}=\frac{2(\sqrt{5}-1)}{4}$$$$y=\frac{\sqrt{5}-1}{2}.sox+y=\sqrt{5}$$

Commented by jagoll last updated on 25/Mar/20

$$\mathrm{great}\mathrm{sir}$$

Answered by $@ty@m123 last updated on 07/Apr/20

$$\frac{\log xy.\log xy}{\log x.\log y}+\frac{\log (x-y).\log (x-y)}{\log x.\log y}=0$$$${\left(\log xy\right)}^2+{\left\{\log (x-y)\right\}}^2=0$$$$\Rightarrow \log xy=0\mid \log (x-y)=0$$$$\Rightarrow xy=1\mid x-y=1$$$$\Rightarrow x=\frac{1}{y}\mid x-y=1$$$$\frac{1}{y}-y=1\mid x=y+1$$$$\Rightarrow y=\frac{\sqrt{5}-1}{2}\mid x=\frac{\sqrt{5}+1}{2}$$$$x+y=\sqrt{5}$$

Commented by jagoll last updated on 25/Mar/20

$$\mathrm{thank}\mathrm{you}\mathrm{sir}$$

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