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Question Number 65446 by LPM last updated on 30/Jul/19

$${10}^{\mathrm{x}}={\mathrm{x}}^{1000}\Rightarrow \mathrm{x}=?$$

Commented by MJS last updated on 30/Jul/19

$$a^x=x^b$$$$x\ln a=b\ln x$$$$x={\mathrm{e}}^{-t}\iff t=-\ln x$$$${\mathrm{e}}^{-t}\ln a=-bt$$$$t{\mathrm{e}}^t=-\frac{\ln a}{b}$$$$\mathrm{now}\mathrm{use}\mathrm{Lambert}\mathrm{W}-\mathrm{function}$$$$\mathrm{in}\mathrm{this}\mathrm{case}$$$$t\approx -8.17477617\vee t\approx -.00230790538$$$$x\approx 3550.26018\vee x\approx 1.00231057$$

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