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This was so satisfying
Multumesc ! 👏
You are welcome 🤗
There are not 32 roots,only 3😅. 2 not found.
a^32 = 2^aln(a^32) = ln(2^a)32*ln|a| = a*ln(2) ===> two cases1st case: a > 032*ln(a) = a*ln(2)ln(a)*a^(-1) = ln(2)/32ln(a)*(e^ln(a))^(-1) = ln(2)/32ln(a)*e^(-ln(a)) = ln(2)/32-ln(a)*e^(-ln(a)) = -ln(2)/32W(-ln(a)*e^(-ln(a))) = W(-ln(2)/32)-ln(a) = W(-ln(2)/32)ln(a) = -W(-ln(2)/32)a = e^(-W(-ln(2)/32)) ===> -1/e < -ln(2)/32 < 0 ===> 2 real solutionsa₁ = e^(-W₀(-ln(2)/32)) = 1.0223929402057803206527516798494005683768365119132864517728278977...a₂ = e^(-W₋₁(-ln(2)/32)) = 256 2nd case: a < 032*ln(-a) = a*ln(2)ln(-a)*a^(-1) = ln(2)/32-ln(-a)*a^(-1) = -ln(2)/32ln(-a)*(-a)^(-1) = -ln(2)/32ln(-a)*(e^ln(-a))^(-1) = -ln(2)/32ln(-a)*e^(-ln(-a)) = -ln(2)/32-ln(-a)*e^(-ln(-a)) = ln(2)/32W(-ln(-a)*e^(-ln(-a))) = W(ln(2)/32)-ln(-a) = W(ln(2)/32)ln(-a) = -W(ln(2)/32)-a = e^(-W(ln(2)/32))a = -e^(-W(ln(2)/32)) ===> ln(2)/32 > 0 ===> 1 real solutiona₃ = -e^(-W₀(ln(2)/32)) = -0.979016934957784612322582550011650068748090048886011676265377083...
❤❤❤❤❤
Thanks ❤️
Nice! For a while, I thought you were going to find all 32 roots. That might take some time lol
La resolví por W Lambert, finalmente me quedó una expresión así: W(-Ln(a)*e^(-Ln(a))=W(-Ln(2)/32)), dando como resultado -Ln(a)=-0.0221458994977...., de donde a=e^0.0221458994977...., a=1.02239, y cumple con la igualdad dada
a^2^5 a^2^2^3 a^2^1^1 a^2^1 (a ➖ 2a+1). 2^(5)=32 2^5 2^2^3 2^1^1 2^1 (a ➖ 2a+1).
32*lna=a*ln2 , lna/a=ln2/2^5 , lna*e^(-lna)=2^3*ln2/2^8 , /*(-1) , W(-lna*e^(-lna))=W(-8*ln2*e^(-8*ln2)) , -lna=-8*ln2 , lna=ln256 , a1=256 , test , a1^32=~ 1.15792*10^155 , 2^a1=~ 1.15792*10^155 , OK , W(-ln2/32)=--lna , a=e^(-W(-ln2/32)) , a2=~ 1.02239 , test , a2^32=~ 2.03129 , 2^a2=~ 2.03129 , OK ,
This was so satisfying
Multumesc ! 👏
You are welcome 🤗
There are not 32 roots,only 3😅. 2 not found.
a^32 = 2^a
ln(a^32) = ln(2^a)
32*ln|a| = a*ln(2) ===> two cases
1st case: a > 0
32*ln(a) = a*ln(2)
ln(a)*a^(-1) = ln(2)/32
ln(a)*(e^ln(a))^(-1) = ln(2)/32
ln(a)*e^(-ln(a)) = ln(2)/32
-ln(a)*e^(-ln(a)) = -ln(2)/32
W(-ln(a)*e^(-ln(a))) = W(-ln(2)/32)
-ln(a) = W(-ln(2)/32)
ln(a) = -W(-ln(2)/32)
a = e^(-W(-ln(2)/32)) ===> -1/e < -ln(2)/32 < 0 ===> 2 real solutions
a₁ = e^(-W₀(-ln(2)/32)) = 1.0223929402057803206527516798494005683768365119132864517728278977...
a₂ = e^(-W₋₁(-ln(2)/32)) = 256
2nd case: a < 0
32*ln(-a) = a*ln(2)
ln(-a)*a^(-1) = ln(2)/32
-ln(-a)*a^(-1) = -ln(2)/32
ln(-a)*(-a)^(-1) = -ln(2)/32
ln(-a)*(e^ln(-a))^(-1) = -ln(2)/32
ln(-a)*e^(-ln(-a)) = -ln(2)/32
-ln(-a)*e^(-ln(-a)) = ln(2)/32
W(-ln(-a)*e^(-ln(-a))) = W(ln(2)/32)
-ln(-a) = W(ln(2)/32)
ln(-a) = -W(ln(2)/32)
-a = e^(-W(ln(2)/32))
a = -e^(-W(ln(2)/32)) ===> ln(2)/32 > 0 ===> 1 real solution
a₃ = -e^(-W₀(ln(2)/32)) = -0.979016934957784612322582550011650068748090048886011676265377083...
❤❤❤❤❤
Thanks ❤️
Nice! For a while, I thought you were going to find all 32 roots. That might take some time lol
La resolví por W Lambert, finalmente me quedó una expresión así: W(-Ln(a)*e^(-Ln(a))=W(-Ln(2)/32)), dando como resultado -Ln(a)=-0.0221458994977...., de donde a=e^0.0221458994977...., a=1.02239, y cumple con la igualdad dada
a^2^5 a^2^2^3 a^2^1^1 a^2^1 (a ➖ 2a+1). 2^(5)=32 2^5 2^2^3 2^1^1 2^1 (a ➖ 2a+1).
32*lna=a*ln2 , lna/a=ln2/2^5 , lna*e^(-lna)=2^3*ln2/2^8 , /*(-1) , W(-lna*e^(-lna))=W(-8*ln2*e^(-8*ln2)) ,
-lna=-8*ln2 , lna=ln256 , a1=256 , test , a1^32=~ 1.15792*10^155 , 2^a1=~ 1.15792*10^155 , OK ,
W(-ln2/32)=--lna , a=e^(-W(-ln2/32)) , a2=~ 1.02239 , test , a2^32=~ 2.03129 , 2^a2=~ 2.03129 , OK ,