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2-x-4x-x-2-x-1-4-x-e-ln-2-x-1-4-xln-2-e-ln-2x-1-4-ln-2-W-ln-2-x-e-ln-2-x-W-1-4-ln-2-ln-2-x-W-1




Question Number 185075 by aba last updated on 16/Jan/23
  2^x =4x ⇒  (x/2^x )=(1/4)                      ⇒x.e^(−ln(2^x )) =(1/4)               ⇒−xln(2)e^(−ln(2x)) =−(1/4)ln(2)              ⇒W(−ln(2^x )e^(−ln(2^x )) )=W(−(1/4)ln(2))              ⇒−ln(2^x )=W(−(1/4)ln(2))             ⇒x=−((W(−(1/4)ln(2))  )/(ln(2))) =((W(−(1/2^4 )ln(2^4 )))/(ln(2)))                        ⇒x=−((W(−ln(2^4 )e^(−ln(2^4 )) ))/(ln(2)))                   ⇒x=((ln(2^4 ))/(ln(2)))             ⇒x=4
$$ \\ $$$$\mathrm{2}^{\mathrm{x}} =\mathrm{4x}\:\Rightarrow\:\:\frac{\mathrm{x}}{\mathrm{2}^{\mathrm{x}} }=\frac{\mathrm{1}}{\mathrm{4}}\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\mathrm{x}.\mathrm{e}^{−\mathrm{ln}\left(\mathrm{2}^{\mathrm{x}} \right)} =\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow−\mathrm{xln}\left(\mathrm{2}\right)\mathrm{e}^{−\mathrm{ln}\left(\mathrm{2x}\right)} =−\frac{\mathrm{1}}{\mathrm{4}}\mathrm{ln}\left(\mathrm{2}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\mathrm{W}\left(−\mathrm{ln}\left(\mathrm{2}^{\mathrm{x}} \right)\mathrm{e}^{−\mathrm{ln}\left(\mathrm{2}^{\mathrm{x}} \right)} \right)=\mathrm{W}\left(−\frac{\mathrm{1}}{\mathrm{4}}\mathrm{ln}\left(\mathrm{2}\right)\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow−\mathrm{ln}\left(\mathrm{2}^{\mathrm{x}} \right)=\mathrm{W}\left(−\frac{\mathrm{1}}{\mathrm{4}}\mathrm{ln}\left(\mathrm{2}\right)\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\mathrm{x}=−\frac{\mathrm{W}\left(−\frac{\mathrm{1}}{\mathrm{4}}\mathrm{ln}\left(\mathrm{2}\right)\right)\:\:}{\mathrm{ln}\left(\mathrm{2}\right)}\:=\frac{\mathrm{W}\left(−\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{4}} }\mathrm{ln}\left(\mathrm{2}^{\mathrm{4}} \right)\right)}{\mathrm{ln}\left(\mathrm{2}\right)} \\ $$$$\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\mathrm{x}=−\frac{\mathrm{W}\left(−\mathrm{ln}\left(\mathrm{2}^{\mathrm{4}} \right)\mathrm{e}^{−\mathrm{ln}\left(\mathrm{2}^{\mathrm{4}} \right)} \right)}{\mathrm{ln}\left(\mathrm{2}\right)}\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\mathrm{x}=\frac{\mathrm{ln}\left(\mathrm{2}^{\mathrm{4}} \right)}{\mathrm{ln}\left(\mathrm{2}\right)} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\mathrm{x}=\mathrm{4} \\ $$
Commented by Frix last updated on 16/Jan/23
I gave the answer in question 184989
$$\mathrm{I}\:\mathrm{gave}\:\mathrm{the}\:\mathrm{answer}\:\mathrm{in}\:\mathrm{question}\:\mathrm{184989} \\ $$
Commented by aba last updated on 16/Jan/23
yes i know
$$\mathrm{yes}\:\mathrm{i}\:\mathrm{know} \\ $$
Commented by Frix last updated on 16/Jan/23
Then where′s the 2^(nd)  solution?
$$\mathrm{Then}\:\mathrm{where}'\mathrm{s}\:\mathrm{the}\:\mathrm{2}^{\mathrm{nd}} \:\mathrm{solution}? \\ $$
Commented by aba last updated on 16/Jan/23
?!
$$?! \\ $$
Commented by Frix last updated on 16/Jan/23
x≈.309907
$${x}\approx.\mathrm{309907} \\ $$

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