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x-2-2-x-solve-for-x-




Question Number 162088 by mathlove last updated on 26/Dec/21
x^2 =2^x   solve   for    x=?
x2=2xsolveforx=?
Answered by Ar Brandon last updated on 26/Dec/21
x^2 =2^x   2lnx=xln2  (1/x)lnx=(1/2)ln2  −lnx∙e^(−lnx) =−((ln2)/2)  W(−lnx)=−((ln2)/2)  −lnx=W_0 (−((ln2)/2))  x=e^(−W_0 (−((ln2)/2)))
x2=2x2lnx=xln21xlnx=12ln2lnxelnx=ln22W(lnx)=ln22lnx=W0(ln22)x=eW0(ln22)
Commented by mathlove last updated on 26/Dec/21
  The numerical answer is
The numerical answer is
Commented by mr W last updated on 26/Dec/21
W(−((ln 2)/2)) has two values  W(−((ln 2)/2))= { ((−1.3862944)),((−0.6931472)) :}  therefore there are two solutions  x= { ((e^(1.3862944) =4)),((e^(0.6931472) =2)) :}
W(ln22)hastwovaluesW(ln22)={1.38629440.6931472thereforetherearetwosolutionsx={e1.3862944=4e0.6931472=2
Commented by Ar Brandon last updated on 26/Dec/21
Thank you, Sir.
Thankyou,Sir.
Commented by Ar Brandon last updated on 26/Dec/21
How to get the 2 values? I got only 0.69...  using the series expansion.
Howtogetthe2values?Igotonly0.69usingtheseriesexpansion.
Commented by mr W last updated on 27/Dec/21
W(−((ln 2)/2)) is the root(s) of  xe^x =−((ln 2)/2)  there are two roots. i got the  numeric values using Grapher:
W(ln22)istheroot(s)ofxex=ln22therearetworoots.igotthenumericvaluesusingGrapher:
Commented by mathlove last updated on 27/Dec/21
thanks   sir
thankssir
Commented by mr W last updated on 27/Dec/21
Answered by Rasheed.Sindhi last updated on 27/Dec/21
Comparing exponents on both sides  and comparing bases on both sides  it′s obvious that x=2.   x⇆2  So x=2
Comparingexponentsonbothsidesandcomparingbasesonbothsidesitsobviousthatx=2.x2Sox=2
Commented by mr W last updated on 26/Dec/21
2^2 =2^2  ⇒x=2  and  4^2 =2^4  ⇒x=4
22=22x=2and42=24x=4

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