AI Claude 3.5 Sonnet would pass the exam. We start with the equation: 2^x + x = 8 Rearrange to: 2^x = 8 - x Multiply both sides by 2^(-x): 1 = (8 - x) * 2^(-x) Multiply both sides by -1/8: -1/8 = (x/8 - 1) * 2^(-x) Let y = x/8 - 1, so x = 8(y + 1) Substituting: -1/8 = y * 2^(-8(y+1)) Multiply both sides by -8: 1 = -8y * 2^(-8(y+1)) 1 = -8y * (1/2^8)^(y+1) 1 = -8y * (1/256)^(y+1) This is in the form of z = we^w, where: z = -1/(256ln(2)) w = -8yln(2) The solution to z = we^w is w = W(z), where W is the Lambert W function So: -8yln(2) = W(-1/(256ln(2))) Solve for y: y = -W(-1/(256ln(2)))/(8ln(2)) Remember that y = x/8 - 1, so: x/8 - 1 = -W(-1/(256ln(2)))/(8ln(2)) Solve for x: x = 8 - W(-1/(256ln(2)))/ln(2) x = 8 - W(1/(256ln(2)))/ln(2) (using the property W(-x) = -W(x)) Simplify: x = 8 - W(256*ln(2))/ln(2)
How do you find out what the lambert function of a constant is? I don’t have W on my calculator.
Wolfram Alpha.
Between 2 & 3, about 2.47
AI Claude 3.5 Sonnet would pass the exam.
We start with the equation: 2^x + x = 8
Rearrange to: 2^x = 8 - x
Multiply both sides by 2^(-x):
1 = (8 - x) * 2^(-x)
Multiply both sides by -1/8:
-1/8 = (x/8 - 1) * 2^(-x)
Let y = x/8 - 1, so x = 8(y + 1)
Substituting:
-1/8 = y * 2^(-8(y+1))
Multiply both sides by -8:
1 = -8y * 2^(-8(y+1))
1 = -8y * (1/2^8)^(y+1)
1 = -8y * (1/256)^(y+1)
This is in the form of z = we^w, where:
z = -1/(256ln(2))
w = -8yln(2)
The solution to z = we^w is w = W(z), where W is the Lambert W function
So: -8yln(2) = W(-1/(256ln(2)))
Solve for y:
y = -W(-1/(256ln(2)))/(8ln(2))
Remember that y = x/8 - 1, so:
x/8 - 1 = -W(-1/(256ln(2)))/(8ln(2))
Solve for x:
x = 8 - W(-1/(256ln(2)))/ln(2)
x = 8 - W(1/(256ln(2)))/ln(2) (using the property W(-x) = -W(x))
Simplify:
x = 8 - W(256*ln(2))/ln(2)
😮
impressionnant j'apprécie fortement ton travail !
a^x + x = b, x = b - W(a^b × ln(a))/ln(a), a ≠ 1, a & b > 0, x > 0.
{2x+2x ➖ }+{x +x ➖ }={4x^2+x^2}=4x^4 2^2x^2^2 1^1x^1^2 x^1^2 (x ➖ 2x+1).
...AND MAYBE IT'D BE BETTER TO MAKE DIFFERENTIAL
........BECAUSE THIS W IS SUCH A GIMMICK
My 10 second guess was somewhere between 2 and 3 so I guessed around 2.5 done.
It is more easy to solve by using graph
Jeez that W constant 😄. Thanks for the lesson
No i could not pass the entrance exam, but who gives a damn ?
Bond.... James Bond ln2
the answer is 2.468
Thank you Einstein.
Legal a funcão W de lambert
you speak too fast
What a nonsense, return me my 11 minutes please...