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Question Number 6045 by sanusihammed last updated on 10/Jun/16

4^x  = 2x    find x

$$\mathrm{4}^{{x}} \:=\:\mathrm{2}{x} \\ $$$$ \\ $$$${find}\:{x} \\ $$

Commented by Yozzii last updated on 10/Jun/16

4^x =2x  2^(2x) =2x  −1=−2x(2^(−2x) )  −1=−2xe^(ln2^(−2x) )   −1=−2xe^(−2xln2)   −ln2=(−2xln2)e^(−2xln2)   ⇒W(−ln2)=−2xln2  x=−((W(−ln2))/(2ln2))    (Following Prakash′s example)

$$\mathrm{4}^{{x}} =\mathrm{2}{x} \\ $$$$\mathrm{2}^{\mathrm{2}{x}} =\mathrm{2}{x} \\ $$$$−\mathrm{1}=−\mathrm{2}{x}\left(\mathrm{2}^{−\mathrm{2}{x}} \right) \\ $$$$−\mathrm{1}=−\mathrm{2}{xe}^{{ln}\mathrm{2}^{−\mathrm{2}{x}} } \\ $$$$−\mathrm{1}=−\mathrm{2}{xe}^{−\mathrm{2}{xln}\mathrm{2}} \\ $$$$−{ln}\mathrm{2}=\left(−\mathrm{2}{xln}\mathrm{2}\right){e}^{−\mathrm{2}{xln}\mathrm{2}} \\ $$$$\Rightarrow{W}\left(−{ln}\mathrm{2}\right)=−\mathrm{2}{xln}\mathrm{2} \\ $$$${x}=−\frac{{W}\left(−{ln}\mathrm{2}\right)}{\mathrm{2}{ln}\mathrm{2}}\:\:\:\:\left({Following}\:{Prakash}'{s}\:{example}\right) \\ $$$$ \\ $$

Answered by Yozzii last updated on 11/Jun/16

x=−((W(−ln2))/(2ln2))

$${x}=−\frac{{W}\left(−{ln}\mathrm{2}\right)}{\mathrm{2}{ln}\mathrm{2}} \\ $$

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