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Question Number 82821 by VBash last updated on 24/Feb/20

$${Log}_{y}x+{Log}_{x}y=64$$$$findxandy$$

Commented by MJS last updated on 24/Feb/20

$$\mathrm{one}\mathrm{equation}\mathrm{in}\mathrm{two}\mathrm{unknown}\mathrm{gives}\mathrm{a}$$$$\mathrm{parametric}\mathrm{solution}$$$$\frac{\ln x}{\ln y}+\frac{\ln y}{\ln x}=64;x>0\wedge y>0$$$$\Rightarrow$$$${\left(\ln x\right)}^2-\left(64\ln y\right)\ln x+{\left(\ln y\right)}^2=0$$$$\Rightarrow$$$$\ln x=(32\pm \sqrt{1023})\ln y$$$$\Rightarrow$$$$x=y^{32\pm \sqrt{1023}}\mathrm{\forall }y>0$$

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