Mathematics
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Logic007:
Math question testing
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Logic007:
\(2^x=2x+1\)
Ultrilliam:
@Angle @Vocaloid
Logic007:
T_T
Ultrilliam:
whats with the face v.v
Logic007:
Just testing,haha...
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Logic007:
solve that eqn using Lambert W function
Logic007:
\(2^x=2x+1\)
Logic007:
\(p^x=ax+b\)
Logic007:
\(-t=x+\frac{b}{a}\)
Logic007:
\(p^{-t-\frac{b}{a}}=a(-t-\frac{b}{a})+b\)
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Logic007:
\(p^{-t-\frac{b}{a}}=-at-b+b\)
Logic007:
\(p^{-t-\frac{b}{a}}=-at\)
Logic007:
\(p^{-t}*p^{-\frac{b}{a}}=-at\)
Logic007:
\(\frac{1}{p^t}*p^{-\frac{b}{a}}=-at\)
Logic007:
\(p^{-\frac{b}{a}}=-at*p^t\)
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Logic007:
\(tp^t=-\frac{1}{a}p^{-\frac{b}{a}}\)
Logic007:
Hmm,is R a complex numberr?
Logic007:
@nuts
nuts:
too lazy to figure out sorry gl fren
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Logic007:
okay,bro
Logic007:
hmm,let me try again
\(tp^t=R\)
Logic007:
\(t=\frac{R}{p^t}\)
Logic007:
\(t=R(e^{-In(p)t})\)
Logic007:
\(t(\frac{In(p)}{R})=In(p)(e^{-In(p)t})\)
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Logic007:
\(t(\frac{In(p)}{R}=In(p)\)
Logic007:
\(t(In(p)=RIn(p)\)
Logic007:
\(RIn(p)=t(In(p))\)
Logic007:
\(W(RIn(p))=t(In(p))\)
Logic007:
\(t=\frac{W(RIn(p))}{In(p)}\)
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Logic007:
Lel,curios
what is \(R\) ? X'D
Logic007:
\(-t=x+\frac{b}{a}\)
\(-t-\frac{b}{a}=x\)
Logic007:
\(x=-t-\frac{b}{a}\)
Logic007:
\(x=-\frac{W(RIn(p))}{In(p)}-\frac{b}{a}\)
Logic007:
\(x=-\frac{W(-\frac{In(p)}{a}p^{-\frac{b}{a}})}{In(p)}-\frac{b}{a}\)
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Logic007:
cool,now I need to apply this concept into the question
Logic007:
\(2^x=2x+1\)
Logic007:
\(-t=x+\frac{1}{2}\)
Logic007:
\(2^{-t-\frac{1}{2}}=2(-t-\frac{1}{2})+1\)
Logic007:
\(2^{-t-\frac{1}{2}}=-2t-1+1\)
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Logic007:
\(2^{-t-\frac{1}{2}}=-2t\)
Logic007:
\(2^{-t}*2^{-\frac{1}{2}}=-2t\)
Logic007:
\(\frac{1}{2^t}*2^{-\frac{1}{2}}=-2t\)
Logic007:
\(2^{-\frac{1}{2}}=-2t*2^t\)
Logic007:
\(t2^t=-\frac{1}{2}*2^{-\frac{1}{2}}\)
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Logic007:
lol,Im stuck coz of that \(R\) T_T
Logic007:
Owh,wait a minute...
lmao,I think I know what is R
R should be a real number if not mistaken
Logic007:
\(t2^t=\frac{1}{2}\)
Logic007:
\(2^t=\frac{1}{2}t\)
Logic007:
\(1=\frac{1}{2}t*2^{-t}\)
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Logic007:
\(1=\frac{1}{2}t*e^{-tIn(2)}\)
Logic007:
\(2=t*e^{-tIn(2)}\)
Logic007:
\(-2In(2)=-tIn(2)*e^{-tIn(2)}\)
Logic007:
\(W(-2In(2))=-tIn(2)\)
Logic007:
\(t=-\frac{W(-2In(2))}{In(2)}\)
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Logic007:
Hmm,something is wrong here
Logic007:
\(1=-2^\frac{3}{2}t*2^t\)
Logic007:
\(1=-2^\frac{3}{2}t*e^{tIn(2)}\)
Logic007:
\(-\frac{In(2)}{2^\frac{3}{2}}=In(2)t*e^{tIn(2)}\)
Logic007:
\(W(-\frac{In(2)}{2^{\frac{3}{2}}})=In(2)t\)
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Logic007:
\(t=\frac{W(\frac{In(2)}{2^{\frac{3}{2}}})}{In(2)}\)
Logic007:
\(R=-2^{-\frac{3}{2}}=-\frac{1}{2}*2^{-\frac{1}{2}}\)
Logic007:
\(-t=x+\frac{b}{a}\)
\(x=-t-\frac{b}{a}\)
\(x=-\frac{W(\frac{In(2)}{2^{\frac{3}{2}}})}{In(2)}-\frac{b}{a}\)
Logic007:
\(x=-\frac{W(-\frac{In(2)}{2}*2^{-\frac{1}{2}})}{In(2)}-\frac{b}{a}\)