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find-x-x-8-x-




Question Number 12446 by tawa last updated on 22/Apr/17
find x  (√x)  =  8^x
findxx=8x
Commented by mrW1 last updated on 23/Apr/17
For equation (√x)=a^x  the solution is  x=−((W(−2ln a))/(2ln a))  where 2ln a≤(1/e) or  a≤e^(1/(2e)) ≈1.202    for a=8>1.202 ⇒ there is no solution.
Forequationx=axthesolutionisx=W(2lna)2lnawhere2lna1eorae12e1.202fora=8>1.202thereisnosolution.
Answered by mrW1 last updated on 23/Apr/17
x≥0  x=^! 8^(2x) =64^x     f(x)=x  g(x)=64^x     f(0)=0  g(0)=1>f(0)    f′(x)=1  g′(x)=64^x ×ln 64>1=f′(x)    ⇒for all x≥0 we have g(x)>f(x)  i.e.  there is no solution for f(x)=g(x)!
x0x=!82x=64xf(x)=xg(x)=64xf(0)=0g(0)=1>f(0)f(x)=1g(x)=64x×ln64>1=f(x)forallx0wehaveg(x)>f(x)i.e.thereisnosolutionforf(x)=g(x)!
Commented by mrW1 last updated on 22/Apr/17
Commented by tawa last updated on 23/Apr/17
God bless you sir
Godblessyousir
Commented by geovane10math last updated on 23/Apr/17
The solution ∉ R  ∈ C
ThesolutionRC
Answered by geovane10math last updated on 22/Apr/17
x = 8^(2x)   x = 64^x   x = e^(x∙ln 64)   (x/e^(x∙ln 64) ) = 1  x∙e^(−x∙ln 64)  = 1  (−ln 64)(x∙e^(−ln 64∙x) ) = −ln 64  −ln 64∙x∙e^(−ln 64∙x)  = −ln 64  −ln 64∙x = y  ye^y  = −ln 64  y = W(−ln 64)    W : Lambert function  x = − ((W(−ln 64))/(ln 64))
x=82xx=64xx=exln64xexln64=1xexln64=1(ln64)(xeln64x)=ln64ln64xeln64x=ln64ln64x=yyey=ln64y=W(ln64)W:Lambertfunctionx=W(ln64)ln64
Commented by mrW1 last updated on 22/Apr/17
W(−ln 64) is not valid!     For Lambert function W(t)  t must be ≥−(1/e) ≈−0.368  but −ln 64≈−4.159 !!!
W(ln64)isnotvalid!ForLambertfunctionW(t)tmustbe1e0.368butln644.159!!!
Commented by tawa last updated on 23/Apr/17
God bless you sir.
Godblessyousir.

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