Entrance Exam | A tricky exponential equation

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[ManojkantSamal]

Respected Sir, Good evening. Pls get us the knowledge of "lambert w function "

[paolovolante]

I gave this question to AI Claude 3.5 Sonnet and this was its answer:
"The equation x^x = 25 can be solved analytically using the Lambert W function. Here's the step-by-step solution:
Start with x^x = 25
Take the natural log of both sides: ln(x^x) = ln(25)
Using the properties of logarithms: x * ln(x) = ln(25)
Multiply both sides by 1/x: ln(x) = ln(25) / x
Multiply both sides by -1: -ln(x) = -ln(25) / x
Take e to the power of both sides: e^(-ln(x)) = e^(-ln(25) / x)
Simplify the left side: 1/x = e^(-ln(25) / x)
Take the reciprocal of both sides: x = e^(ln(25) / x)
This is in the form of y = e^(1/y), which has the solution y = e^W(1), where W is the Lambert W function.
In our case, we have x = e^(ln(25) / x), so the solution is: x = e^W(ln(25)) "

[menw]

Does not make sense e^W(ln 25), how do you work this out?

[payoo_]

x^2 = 3^x
ln(x^2) = ln(3^x)
2*ln|x| = x*ln(3) ===> two cases

1st case: x > 0
2*ln(x) = x*ln(3)
ln(x)*x^(-1) = ln(3)/2
ln(x)*e^ln(x^(-1)) = ln(3)/2
ln(x)*e^(-ln(x)) = ln(3)/2
-ln(x)*e^(-ln(x)) = -ln(3)/2
W(-ln(x)*e^(-ln(x))) = W(-ln(3)/2)
-ln(x) = W(-ln(3)/2)
ln(x) = -W(-ln(3)/2)
x = e^(-W(-ln(3)/2)) ===> -ln(3)/2 < -1/e ===> no real solutions

2nd case: x < 0
2*ln(-x) = x*ln(3)
ln(-x)*x^(-1) = ln(3)/2
-ln(-x)*x^(-1) = -ln(3)/2
ln(-x)*(-x)^(-1) = -ln(3)/2
ln(-x)*e^ln((-x)^(-1)) = -ln(3)/2
ln(-x)*e^(-ln(-x)) = -ln(3)/2
-ln(-x)*e^(-ln(-x)) = ln(3)/2
W(-ln(-x)*e^(-ln(-x))) = W(ln(3)/2)
-ln(-x) = W(ln(3)/2)
ln(-x) = -W(ln(3)/2)
-x = e^(-W(ln(3)/2)
x = -e^(-W(ln(3)/2)) ===> ln(3)/2 > 0 ===> 1 real solution
x = -e^(-W₀(ln(3)/2)) =

[Why-kb_]

before watching: xln(x)=ln(25)
ln(x)e^ln(x)=ln(25)
ln(x)=W(ln(25)
x=e^W(2ln(5))

[SALogics]

By ispection 2<x<3
x = 2.95 approximately

[astroide-fsuf]

You showoff the lambert function too much

[justinwooten]

x^x = a, x = e^(W(ln(a))) for x > 0.

[josephcamba]

this is not intuitive. this type of stuff turns most people away from math.

[fahadmshasan]

Please explaine, why in e^lnx, e and ln cancel out??🤔🤔🙏🙏❤❤❤