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Question Number 180260 by Mastermind last updated on 09/Nov/22

$${\mathrm{x}}^{\mathrm{logx}}=100\mathrm{x}$$$$$$$$\mathrm{what}\mathrm{is}\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{x}?$$

Answered by Frix last updated on 09/Nov/22

$$\mathrm{if}\log \mathrm{means}{\log }_{10}$$$$x^{\log x}=100x$$$$\log x^{\log x}=\log 100x$$$${\left(\log x\right)}^2=2+\log x$$$${\left(\log x\right)}^2-\log x-2=0$$$$\log x=\frac{1}{2}\pm \sqrt{\frac{1}{4}+2}=\frac{1}{2}\pm \frac{3}{2}=\{\begin{array}{l}-1\\ 2\end{array}$$$$x=\frac{1}{10}\vee x=100$$

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