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Question Number 17742 by tawa tawa last updated on 10/Jul/17

x^3  = 3^x ,   find x.

x3=3x,findx.

Commented by 1kanika# last updated on 10/Jul/17

x=3

x=3

Answered by mrW1 last updated on 10/Jul/17

x=3^(x/3) =e^((x/3)ln 3)   xe^(−(x/3)ln 3) =1  (−(x/3)ln 3)e^(−(x/3)ln 3) =−((ln 3)/3)  −(x/3)ln 3=W(−((ln 3)/3))  ⇒x=−(3/(ln 3))W(−((ln 3)/3))= { ((2.4...)),(3) :}

x=3x3=ex3ln3xex3ln3=1(x3ln3)ex3ln3=ln33x3ln3=W(ln33)x=3ln3W(ln33)={2.4...3

Commented by tawa tawa last updated on 10/Jul/17

God bless you sir. i understand it. i really appreciate.

Godblessyousir.iunderstandit.ireallyappreciate.

Commented by mrW1 last updated on 10/Jul/17

solve x^2 =3^x

solvex2=3x

Commented by tawa tawa last updated on 10/Jul/17

x^2  = 3^x   3^(x/2)  = x  e^((x/2)ln3 ) = x  1 = xe^(−((xln3)/2))   −((ln3)/2) = (−((xln3)/2))e^((−((xln3)/2)))   −((xln3)/2) = We^((−((ln3)/2)))   x = ((−2We^((−((ln3)/2))) )/(ln3))  x = ((−2We^((((ln3)/2))) )/(ln3))  x = ((− 2 × 0.376839)/(1.09861))  x = ((− 0.753674)/(1.09861))  x = − 0.6860

x2=3x3x2=xex2ln3=x1=xexln32ln32=(xln32)e(xln32)xln32=We(ln32)x=2We(ln32)ln3x=2We(ln32)ln3x=2×0.3768391.09861x=0.7536741.09861x=0.6860

Commented by mrW1 last updated on 10/Jul/17

x^2  = 3^x   ⇒±3^(x/2)  = x  ......  totally 3 solutions

x2=3x±3x2=x......totally3solutions

Commented by tawa tawa last updated on 10/Jul/17

ohh, i see.... it almost affect the last answer

ohh,isee....italmostaffectthelastanswer

Commented by tawa tawa last updated on 10/Jul/17

Sir.. .. x^3  has two solutions  x^2  has three solutions ...  or it depends on the question.

Sir....x3hastwosolutionsx2hasthreesolutions...oritdependsonthequestion.

Commented by tawa tawa last updated on 10/Jul/17

Show me sir.

Showmesir.

Commented by mrW1 last updated on 10/Jul/17

if x^3 =3^x   since 3^x >0  ⇒x^3 >0  ⇒x>0  ⇒x^3 =3^x ≡x=3^(x/3)     if x^2 =3^x   x can also be negative,  ⇒x^2 =3^x ≡x=±3^(x/2)     just like  x^3 =8  ⇒x=(8)^(1/3) =2  x^3 =−8  ⇒x=(−8)^(1/3) =−2  but  x^2 =4  ⇒x=±(4)^(1/2) =±2

ifx3=3xsince3x>0x3>0x>0x3=3xx=3x3ifx2=3xxcanalsobenegative,x2=3xx=±3x2justlikex3=8x=(8)13=2x3=8x=(8)13=2butx2=4x=±(4)12=±2

Commented by tawa tawa last updated on 10/Jul/17

Now, i understand sir. God bless you

Now,iunderstandsir.Godblessyou

Commented by mrW1 last updated on 10/Jul/17

solution for x^4 =3^x   x=±3^(x/4)   x=±e^((x/4)ln 3)   xe^(−(x/4)ln 3) =±1  (−(x/4)ln 3)e^(−(x/4)ln 3) =±((ln 3)/4)  −(x/4)ln 3=W(±((ln 3)/4))  ⇒x=−(4/(ln 3))W(±((ln 3)/4))  x= { ((−(4/(ln 3)) W(((ln 3)/4))=−0.80225)),((−(4/(ln 3)) W(−((ln 3)/4))= { ((1.51687)),((7.17476)) :})) :}

solutionforx4=3xx=±3x4x=±ex4ln3xex4ln3=±1(x4ln3)ex4ln3=±ln34x4ln3=W(±ln34)x=4ln3W(±ln34)x={4ln3W(ln34)=0.802254ln3W(ln34)={1.516877.17476

Commented by tawa tawa last updated on 10/Jul/17

I really appreciate your effort sir. God bless you sir.

Ireallyappreciateyoureffortsir.Godblessyousir.

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