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Mathematics 15 Online
Logic007:

Math question testing

Logic007:

\(2^x=2x+1\)

Ultrilliam:

@Angle @Vocaloid

Logic007:

T_T

Ultrilliam:

whats with the face v.v

Logic007:

Just testing,haha...

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:

\(p^{-t-\frac{b}{a}}=a(-t-\frac{b}{a})+b\)

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:

\(p^{-\frac{b}{a}}=-at*p^t\)

Logic007:

\(tp^t=-\frac{1}{a}p^{-\frac{b}{a}}\)

Logic007:

Found this,lel http://prntscr.com/jscimc where did the \(R\) come from?

Logic007:

Hmm,is R a complex numberr?

Logic007:

@nuts

nuts:

too lazy to figure out sorry gl fren

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:

\(t(\frac{In(p)}{R})=In(p)(e^{-In(p)t})\)

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:

\(t=\frac{W(RIn(p))}{In(p)}\)

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:

\(x=-\frac{W(-\frac{In(p)}{a}p^{-\frac{b}{a}})}{In(p)}-\frac{b}{a}\)

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:

\(2^{-t-\frac{1}{2}}=-2t-1+1\)

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:

\(t2^t=-\frac{1}{2}*2^{-\frac{1}{2}}\)

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:

\(1=\frac{1}{2}t*2^{-t}\)

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:

\(t=-\frac{W(-2In(2))}{In(2)}\)

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:

\(W(-\frac{In(2)}{2^{\frac{3}{2}}})=In(2)t\)

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:

\(x=-\frac{W(-\frac{In(2)}{2}*2^{-\frac{1}{2}})}{In(2)}-\frac{b}{a}\)

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