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Question Number 121899 by ZiYangLee last updated on 12/Nov/20

$$\mathrm{Solve}{\left(\log 2x\right)}^2+{\left(\log 3x\right)}^2={\left(\log 2\right)}^2+{\left(\log 3\right)}^2$$

Commented by Dwaipayan Shikari last updated on 12/Nov/20

$$x=1$$$$2{log}^{2}x+2log2logx+2log3logx=0$$$${log}^{2}x+logxlog6=0$$$$logx=-log6\Rightarrow x=\frac{1}{6}orlogx=0,x=1$$

Commented by ZiYangLee last updated on 12/Nov/20

$$\mathrm{thanks}\mathrm{sir}$$

Answered by MJS_new last updated on 12/Nov/20

$${\left(\log nx\right)}^2={\left(\log x\right)}^2+2\log n\log x+{\left(\ln n\right)}^2$$$$\mathrm{transforming}\mathrm{the}\mathrm{given}\mathrm{equation}\mathrm{we}\mathrm{get}$$$$\log x\log \left(6x\right)=0$$$$\Rightarrow x=1\vee x=\frac{1}{6}$$

Commented by ZiYangLee last updated on 12/Nov/20

$$\mathrm{Thanks}\mathrm{sir}..★$$

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