Menu Close

solve-for-x-R-x-x-n-a-with-n-a-R-find-also-the-range-of-a-such-that-a-solution-exists-




Question Number 139805 by mr W last updated on 01/May/21
solve for x∈R  x^x^n  =a  with n, a∈R^+   find also the range of a such that a  solution exists.
solveforxR\boldsymbolx\boldsymbolx\boldsymboln=\boldsymbolawithn,aR+findalsotherangeof\boldsymbolasuchthatasolutionexists.
Commented by mr W last updated on 01/May/21
i thought about this general equation  for a long time and tried to find  a formula to express the solution.  we know if n=a, the solution for  x^x^a  =a is the same as for x^a =a, i.e.  x=(a)^(1/a) . but what is the solution for  the general case with n≠a?
ithoughtaboutthisgeneralequationforalongtimeandtriedtofindaformulatoexpressthesolution.weknowifn=a,thesolutionforxxa=aisthesameasforxa=a,i.e.x=aa.butwhatisthesolutionforthegeneralcasewithna?
Answered by mr W last updated on 02/May/21
here the solution i got for the  general case x^x^n  =a.  x^x^n  =a  x=a^(1/x^n ) =e^((ln a)/x^n )   x^n =e^((nln a)/x^n )   (1/x^n )e^((nln a)/x^n ) =1  ((nln a)/x^n )e^((nln a)/x^n ) =nln a  ((nln a)/x^n )=W(nln a)  x^n =((nln a)/(W(nln a)))  ⇒x=(((nln a)/(W(nln a))))^(1/n)   such that a real root exists, W(nln a)  must be defined. this is the case if  nln a≥−(1/e), i.e. a≥e^(−(1/(ne))) =(1/( (e)^(1/(ne)) )).  therefore:  for a<(1/( (e)^(1/(ne)) )) ⇒ no root  for a=(1/( (e)^(1/(ne)) )) ⇒ one root  for (1/( (e)^(1/(ne)) ))<a<1 ⇒ two roots  for a≥1 ⇒ one root    example: x^x^5  =10  ⇒x=(((5ln 10)/(W(5ln 10))))^(1/5) ≈(((5ln 10)/(1.835928)))^(1/5) =1.443664    example: x^x^3  =(9/(10))  ⇒x=(((3ln 0.9)/(W(3ln 0.9))))^(1/3) ≈ { (((((3ln 0.9)/(−1.656404)))^(1/3) =0.575719)),(((((3ln 0.9)/(−0.545257)))^(1/3) =0.833808)) :}
herethesolutionigotforthegeneralcasexxn=a.xxn=ax=a1xn=elnaxnxn=enlnaxn1xnenlnaxn=1nlnaxnenlnaxn=nlnanlnaxn=W(nlna)xn=nlnaW(nlna)x=nlnaW(nlna)nsuchthatarealrootexists,W(nlna)mustbedefined.thisisthecaseifnlna1e,i.e.ae1ne=1ene.therefore:fora<1enenorootfora=1eneonerootfor1ene<a<1tworootsfora1onerootexample:xx5=10x=5ln10W(5ln10)55ln101.8359285=1.443664example:xx3=910x=3ln0.9W(3ln0.9)3{3ln0.91.6564043=0.5757193ln0.90.5452573=0.833808
Commented by Tawa11 last updated on 23/Jul/21
great
greatgreat

Leave a Reply

Your email address will not be published. Required fields are marked *