Math Is Fun Forum

iheartmaths

Member

Registered: 2008-01-08

Posts: 31

Iterated Exponential Constants: x^2=2^x

Hi guys and gals,

I posed this problem to myself one night and quickly discovered that it is less innocuous than it seems! Graphically, the solutions are 2, 4, and a negative transcendental number (-0.7666646959...). However, out of curiosity I decided to see if there was an algebraic solution. From what I’ve gathered on the internet, there does not seem to be one. Also, as can be easily seen, the problem has two obvious solutions (2, 4), of which I was only capable of finding x equals 2. Thus, I am convinced that I did something wrong.



I am sorry if my LaTeX is a little rough. I have just learned it through Dross' crash course. It was very confusing to interject text into the mathematics. So, I will explain any ambiguous steps if needed (at least until I get better), but somehow I doubt the need will arise.
Otherwise, would anyone care to comb it over to help me spot the mistake?

Thank you very much in advance.

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Don't quote me on that.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Iterated Exponential Constants: x^2=2^x

I think the mistake is here. Multiplying top and bottom by x^(1/x) would make the denominator read x^(2/x).

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Why did the vector cross the road?
It wanted to be normal.

iheartmaths

Member

Registered: 2008-01-08

Posts: 31

Re: Iterated Exponential Constants: x^2=2^x

I think you are correct, mathsyperson. Thank you. I do tend to make mistakes such as those.

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Don't quote me on that.