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

    10^x  = x^(1000)  ⇒ x=?

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

Commented by MJS last updated on 30/Jul/19

a^x =x^b   xln a =bln x  x=e^(−t)  ⇔ t=−ln x  e^(−t) ln a =−bt  te^t =−((ln a)/b)  now use Lambert W−function  in this case  t≈−8.17477617∨t≈−.00230790538  x≈3550.26018∨x≈1.00231057

$${a}^{{x}} ={x}^{{b}} \\ $$$${x}\mathrm{ln}\:{a}\:={b}\mathrm{ln}\:{x} \\ $$$${x}=\mathrm{e}^{−{t}} \:\Leftrightarrow\:{t}=−\mathrm{ln}\:{x} \\ $$$$\mathrm{e}^{−{t}} \mathrm{ln}\:{a}\:=−{bt} \\ $$$${t}\mathrm{e}^{{t}} =−\frac{\mathrm{ln}\:{a}}{{b}} \\ $$$$\mathrm{now}\:\mathrm{use}\:\mathrm{Lambert}\:\mathrm{W}−\mathrm{function} \\ $$$$\mathrm{in}\:\mathrm{this}\:\mathrm{case} \\ $$$${t}\approx−\mathrm{8}.\mathrm{17477617}\vee{t}\approx−.\mathrm{00230790538} \\ $$$${x}\approx\mathrm{3550}.\mathrm{26018}\vee{x}\approx\mathrm{1}.\mathrm{00231057} \\ $$

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