A great olympiad question - Exponential Equation👍

A great challenge. Math Olympiad Question
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#maths #math #mathematics #algebra

[@hk1351614]

Stop clickbaiting. x=256 is one of the three real solutions, not to mention there's infinite many solutions. Either the title was flawed or the answer you get was flawed, more likely both.

[@Jiyoon02]

I can't agree more. These, so claimed "Math Olympiad Question" channels are getting out of hand and disgustingly annoying. The quality of questions they propose are no olympiad quality, mostly pathetic. Not to mention that they are very often flawed. And worse yet, these guys are too inferior that they don't even know how to solve the questions they themselves proposed. 9 of 10 times resorting to some pathetic and flawed arithmetic child's play. Whenever I see these fake "Olympiad" channels, I feel so much disgusts for their guts to clickbait these as "Olympiad " material. The worst part is that they seem to multiply endlessly. I see channels like these again and again, despite clicking "don't recommend" to each and every one of them whenever I see them.

[@pelayomedina2174]

There is also a solution between 1 and 2

[@JohnSmith-nx7zj]

And a solution between -1 and 0

[@Hassan.Wahba.97]

I solved it by looking at it in less than 10 seconds with this method: assume x is some power of 2: and let it be 2^y. then we have 2 ^(2^y) = 2^y^32 --> 2^y = 32y --> 2^(y-5) = y --> y = 8 --> x = 2^8 = 256

[@cos7uh]

this is literally only guess and check

[@kamilhorvat8290]

Could you also provide a reasoning why we should perform all these operations on the equation? So that the solution is a bit more understandable and palatable. What if the initial equation was slightly different, let's say 3^x=x^23. Would similar sequence of operations lead to solution ?

[@user-rk5xo7yj9y]

i don’t understand the solution at all 😭

[@VasylOstapchuk]

There are 3 real solutions.

[@Rai_Te]

Ok ... you get 5 out of 10 points ... The remaining 5 points would have been granted, if you had found the second solution to the equation ... (x=1.02239294)

[@tschantz]

Whoa!

[@cytos]

You get 10/15 points. The remaining 5 points would have been granted if you had found the third solution to the equation: x ≈ -0.979016934957784612322582550011650068748090048886011676265377083

[@rohan19493]

I dont think the last line was correct

[@TanoFotang]

Eqn can be rewritten as ln(x)/x=ln(2)/32 => -ln(x)*e^(-ln(x))=-ln(2)/32, giving x=e^(-W(-ln(2)/32)). Real values: 256 and 1.02239294020..., W being the product log. I cannot find the 3rd real value given by user @cytos.

[@balthazarbeutelwolf9097]

If you form the function f(x)=2^x-x^32 then the roots of f are the solutions of the equation. Now, f(0)=1 and f(-1)=-0.5, hence - as f is clearly continuous - there is a solution between 0 and -1.