Question Number 65446 by LPM last updated on 30/Jul/19
$$\:\:\:\:\mathrm{10}^{\mathrm{x}} \:=\:\mathrm{x}^{\mathrm{1000}} \:\Rightarrow\:\mathrm{x}=? \\ $$
Commented by MJS last updated on 30/Jul/19
$${a}^{{x}} ={x}^{{b}} \\ $$$${x}\mathrm{ln}\:{a}\:={b}\mathrm{ln}\:{x} \\ $$$${x}=\mathrm{e}^{−{t}} \:\Leftrightarrow\:{t}=−\mathrm{ln}\:{x} \\ $$$$\mathrm{e}^{−{t}} \mathrm{ln}\:{a}\:=−{bt} \\ $$$${t}\mathrm{e}^{{t}} =−\frac{\mathrm{ln}\:{a}}{{b}} \\ $$$$\mathrm{now}\:\mathrm{use}\:\mathrm{Lambert}\:\mathrm{W}−\mathrm{function} \\ $$$$\mathrm{in}\:\mathrm{this}\:\mathrm{case} \\ $$$${t}\approx−\mathrm{8}.\mathrm{17477617}\vee{t}\approx−.\mathrm{00230790538} \\ $$$${x}\approx\mathrm{3550}.\mathrm{26018}\vee{x}\approx\mathrm{1}.\mathrm{00231057} \\ $$