Nice Exponential Problem | Olympiad Math Algebra Question

[@اسماعیلخسروی-خ6ظ]

❤❤❤❤❤

[@MathBeast.channel-l9i]

Thanks ❤️

[@DedMatveev]

There are not 32 roots,only 3😅. 2 not found.

[@payoo_2674]

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...

[@joseeoliviero6078]

Nice! For a while, I thought you were going to find all 32 roots. That might take some time lol

[@Maria-wn4du]

Multumesc ! 👏

[@MathBeast.channel-l9i]

You are welcome 🤗

[@prollysine]

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 ,