What is the solution for x in the equation x[sup]2[/sup] = 2[sup]x[/sup]?
It has two solutions: 2 and 4. Don’t ask me about the math, I just slapped together a quick and dirty program to figure it out.
There’s no nice algebraic way to solve this one, I don’t think. Solution-by-inspection, or maybe a Taylor series expansion of 2[sup]x[/sup] might be the way to take a stab at it.
But you have your answer anyway.
I think it’s got three solutions, but I’m not sure.
It should, actually, on the -x side.
At x=0, 2[sup]x[/sup] is 1, and is still decreasing as you go towards more negative x. However, x[sup]2[/sup] is at 0 and starts increasing again.
A simple binary search’d get the answer fast enough for government work.
Well, I ran my program with a range of -10000 to 10000 and it only spat out 2 and 4. I also ran it with the constant from 2 to 6. Interestingly, out of those, only 2 and 4 had the two solutions 2 and 4, all the others had only only the one solution constant (k) = x.
the third solution is roughly -0.7666.
Yup, got that myself. Damn, you beat me to it. Your code-fu is faster than mine. 
This was just done in the past three months, and someone did link to an algebraic solution. It was fairly clever. No idea who the OP was, but I briefly mentioned Lambert’s W function, and that may help searches.
Out of curiosity, I ran it through MathCad and it spit out this algebraic solution.. Don’t ask me what W is. Can anyone explain this solution?
There was recently a thread on this (which I’m at a loss as to how to search for) where a linked page had a derived solution, and I believe there is a large number (infinite?) of non-integer solutions.
The linked page was one in a series that was something along the lines of “properties of interesting numbers”.
No that was slightly different, then I was looking for solutions to the more generalized x[sup]y[/sup] = y[sup]x[/sup]
Really all I want is the exact surd form for the third solution.
I figured I’d attempt to solve it algebraically even though all 3 roots have been found.
I took the log of both sides which gives us:
2log(x) = xlog(2) then:
2/(log(2)) = x/(log(x)) then:
6.643856190 = x/(log(x))
Well that’s as far as I got.
Anyone care to continue ?
The third solution may well be transcendental, which won’t help you–I think, cause surd’s not a common term in America these days.
Mathworld describes Lambert’s W function, which is what showed up in Q.E.D.'s solution.
Wow !!!
You math “dopers” are fast !!!
I typed in my reply after I read the reply from **William_Ashbless ** and in the meantime, 4 postings went through !!
Here’s the other thread started by MC Master of Ceremonies.
As a cool side note (at least for me), I solved the equation "X^2 = 2^x) with my TI-89 and and it said busy in the corner for about 15 seonds and then gave me this warning message right next to the busy signal “Warning, there may be more solutions” and then it gave me the 3 solutions from above. (2, 4, -7.6)
Anyway, I’ve never had it give me a busy signal for near that long or a warning message like that before…
Are there any perfectly acceptable math problems that have exact answers but cannot be solved analyticaly? If so is there an underlying meaning to this?
What exactly does “transcendental” mean? I see this used all the time but I never know what it means.
There are a great deal of examples for your first question. One of the most remarkable results of Galois theory (a branch of abstract algebra) declared that the general solution of polynomial equations of degree greater than four cannot be expressed using radicals. I suppose that’s what you meant by “solved analytically”.
A real number is called algebraic if it is the root of a polynomial with integer coefficients. A real number that is not algebraic is called transcendental. Interestingly, there are more transcendental numbers than algebraic numbers (since the former set is uncountable, while the latter set is countable).
Yes, very many. Considering that most real numbers don’t have names, all the ones we see exact solutions for are special cases.
I refer you to the works of Chaitin and Gödel.
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A transcendental real is one which is not the root of a polynomial with integer coefficients.