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Question Number 155215 by 0731619 last updated on 27/Sep/21

Answered by MJS_new last updated on 27/Sep/21

$$x=\frac{1}{2}\wedge y=\frac{1}{3}$$

Commented by 0731619 last updated on 27/Sep/21

$$solution???$$

Commented by MJS_new last updated on 27/Sep/21

$$\mathrm{sorry}{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{on}\mathrm{first}\mathrm{sight}$$

Commented by 0731619 last updated on 27/Sep/21

$$plesesolution$$

Answered by Rasheed.Sindhi last updated on 28/Sep/21

$$\underset{\left(\mathrm{i}\right)}{\underbrace{{\left(2x\right)}^{\ln 2}={\left(3\mathrm{y}\right)}^{\ln 3}}}\wedge \underset{\left(\mathrm{ii}\right)}{\underbrace{3^{\mathrm{lnx}}=2^{\mathrm{lny}}}};\mathrm{x}=?$$$$\left(\mathrm{i}\right)\Rightarrow 2^{\ln 2}{\mathrm{x}}^{\ln 2}=3^{\ln 3}{\mathrm{y}}^{\ln 3}$$$$\frac{{\mathrm{x}}^{\ln 2}}{{\mathrm{y}}^{\ln 3}}=\frac{3^{\ln 3}}{2^{\ln 2}}$$$$\ln \left(\frac{{\mathrm{x}}^{\ln 2}}{{\mathrm{y}}^{\ln 3}}\right)=\ln \left(\frac{3^{\ln 3}}{2^{\ln 2}}\right)$$$$\ln \left({\mathrm{x}}^{\ln 2}\right)-\ln \left({\mathrm{y}}^{\ln 3}\right)=\ln \left(3^{\ln 3}\right)-\ln \left(2^{\ln 2}\right)$$$$\ln 2\mathrm{lnx}-\ln 3\mathrm{lny}={\left(\ln 3\right)}^2-{\left(\ln 2\right)}^2...\mathrm{A}$$$$\left(\mathrm{ii}\right)\Rightarrow 3^{\mathrm{lnx}}=2^{\mathrm{lny}}\Rightarrow \mathrm{lnxln}3=\mathrm{lnyln}2$$$$\Rightarrow \mathrm{lny}=\left(\frac{\ln 3}{\ln 2}\right)\mathrm{lnx}...................\mathrm{B}$$$$\mathrm{A}:\ln 2\mathrm{lnx}-\ln 3\mathrm{lny}={\left(\ln 3\right)}^2-{\left(\ln 2\right)}^2$$$$\Rightarrow \ln 2\mathrm{lnx}-\ln 3\left(\left(\frac{\ln 3}{\ln 2}\right)\mathrm{lnx}\right)={\left(\ln 3\right)}^2-{\left(\ln 2\right)}^2$$$$\Rightarrow \mathrm{lnx}(\ln 2-\frac{{\left(\ln 3\right)}^2}{\ln 2})={\left(\ln 3\right)}^2-{\left(\ln 2\right)}^2$$$$\Rightarrow \mathrm{lnx}\left(\frac{{\left(\ln 2\right)}^2-{\left(\ln 3\right)}^2}{\ln 2}\right)={\left(\ln 3\right)}^2-{\left(\ln 2\right)}^2$$$$\Rightarrow -\mathrm{lnx}\left(\frac{{\left(\ln 3\right)}^2-{\left(\ln 2\right)}^2}{\ln 2}\right)={\left(\ln 3\right)}^2-{\left(\ln 2\right)}^2$$$$\frac{-\mathrm{lnx}}{\ln 2}=1\Rightarrow \mathrm{lnx}=-\ln 2$$$$\mathrm{lnx}={\ln 2}^{-1}$$$$\mathrm{x}=2^{-1}=\frac{1}{2}$$$$\mathrm{B}:\mathrm{lny}=\left(\frac{\ln 3}{\ln 2}\right)\mathrm{lnx}\Rightarrow \mathrm{lny}=\left(\frac{\ln 3}{\ln 2}\right){\ln 2}^{-1}$$$$\mathrm{lny}=-\left(\frac{\ln 3}{\ln 2}\right)\ln 2={\ln 3}^{-1}$$$$\mathrm{y}=3^{-1}=\frac{1}{3}$$

Commented by Rasheed.Sindhi last updated on 28/Sep/21

$$\mathcal{T}heansweriscompletenow.$$

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