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Question Number 65333 by aliesam last updated on 28/Jul/19

Commented by Prithwish sen last updated on 28/Jul/19

$${\log }_{\mathrm{y}}\mathrm{x}=\mathrm{v}\Rightarrow \frac{{\mathrm{v}}^2+1}{\mathrm{v}}=\frac{17}{4}\Rightarrow {4\mathrm{v}}^2-17\mathrm{v}+4=0$$$${4\mathrm{v}}^2-16\mathrm{v}-\mathrm{v}+4=0\Rightarrow 4\mathrm{v}(\mathrm{v}-4)-1(\mathrm{v}-4)=0$$$$\mathrm{v}=\frac{1}{4},4\Rightarrow \mathrm{y}={\mathrm{x}}^4\mathrm{or}{\mathrm{y}}^4=\mathrm{x}$$$$\mathbf{when}\mathbf{y}={\mathbf{x}}^4\mathbf{then}{\mathrm{x}}^5=3^5\therefore \mathrm{x}=3\mathrm{and}\mathrm{y}=81$$$$\mathbf{when}{\mathbf{y}}^4=\mathbf{x}\mathbf{then}\mathrm{x}=81\mathrm{and}\mathrm{y}=3$$$$\mathrm{please}\mathrm{check}.$$

Answered by Tanmay chaudhury last updated on 28/Jul/19

$${log}_{y}x+{log}_{x}y=\frac{17}{4}$$$$letlogx=alogy=b$$$$\frac{a}{b}+\frac{b}{a}=\frac{17}{4}$$$$4a^2+4b^2-17ab=0$$$$4a^2-16ab-ab+4b^2=0$$$$4a(a-4b)-b(a-4b)=0$$$$(a-4b)(4a-b)=0$$$$a=4bora=\frac{b}{4}$$$$whena=4b$$$$logx=4logy$$$$x=y^4$$$$\left(xy\right)=243$$$$\left(y^5\right)=243=3^5$$$$y=3andx=3^4=81$$$$whena=\frac{b}{4}$$$$logx=\frac{logy}{4}$$$$y=x^4$$$$\left(xy\right)=243$$$$\left(x^{}\times x^4\right)=243$$$$x^5=3^5$$$$x=3andy=3^4=81$$$$\mathit{s}\mathit{o}\mathit{w}\mathit{h}\mathit{e}\mathit{n}\mathit{x}=3\mathit{y}=81$$$$\mathit{a}\mathit{n}\mathit{d}\mathit{w}\mathit{h}\mathit{e}\mathit{n}\mathit{x}=81\mathit{y}=3$$$$$$

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