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Question Number 45500 by Sanjarbek last updated on 13/Oct/18

Answered by MrW3 last updated on 13/Oct/18

x=64^(1/x) =e^((ln 64)/x) 1=(1/x)e^((ln 64)/x) ln 64=((ln 64)/x)e^((ln 64)/x) ((ln 64)/x)=W(ln 64) ←Lambert W function ⇒x=((ln 64)/(W(ln 64)))=((ln 64)/(1.223517))=3.3991

$${x}=\mathrm{64}^{\frac{\mathrm{1}}{{x}}} ={e}^{\frac{\mathrm{ln}\:\mathrm{64}}{{x}}} \\ $$$$\mathrm{1}=\frac{\mathrm{1}}{{x}}{e}^{\frac{\mathrm{ln}\:\mathrm{64}}{{x}}} \\ $$$$\mathrm{ln}\:\mathrm{64}=\frac{\mathrm{ln}\:\mathrm{64}}{{x}}{e}^{\frac{\mathrm{ln}\:\mathrm{64}}{{x}}} \\ $$$$\frac{\mathrm{ln}\:\mathrm{64}}{{x}}=\mathbb{W}\left(\mathrm{ln}\:\mathrm{64}\right)\:\leftarrow{Lambert}\:{W}\:{function} \\ $$$$\Rightarrow{x}=\frac{\mathrm{ln}\:\mathrm{64}}{\mathbb{W}\left(\mathrm{ln}\:\mathrm{64}\right)}=\frac{\mathrm{ln}\:\mathrm{64}}{\mathrm{1}.\mathrm{223517}}=\mathrm{3}.\mathrm{3991} \\ $$

Commented by Cheyboy last updated on 14/Oct/18

Sir Mr W_3 how can i have acess to Lambert function calculator

$$\mathrm{Sir}\:\mathrm{Mr}\:\mathrm{W}_{\mathrm{3}} \:\mathrm{how}\:\mathrm{can}\:\mathrm{i}\:\mathrm{have}\:\mathrm{acess} \\ $$$$\mathrm{to}\:\mathrm{Lambert}\:\mathrm{function}\:\mathrm{calculator} \\ $$

Commented by Joel578 last updated on 14/Oct/18

wolframalpha.com

$$\mathrm{wolframalpha}.\mathrm{com} \\ $$

Commented by Cheyboy last updated on 14/Oct/18

Ok thank u sir

$$\mathrm{Ok}\:\mathrm{thank}\:\mathrm{u}\:\mathrm{sir} \\ $$

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