Solved or unsolved?

Discus: SAT/ACT Tests and Test Preparation: July 2004 Archive: Solved or unsolved?
 By Tongos (Tongos) on Sunday, July 11, 2004 - 06:43 pm: Edit

does anybody know a short way to solve x^x=5 or does anybody have a way?

 By Al0 (Al0) on Sunday, July 11, 2004 - 08:46 pm: Edit

Somebody mentioned Newton's:
x^x=5
0=x^x-5
f(x)=x^x-5
then to take the derivative:
f(x)=e^(ln(x^x))-5
f(x)=e^(x*ln(x))-5
f'(x)=(x*(1/x)+1*ln(x))*x^x-5
f'(x)=(ln(x)+1)*x^x-5

The x-intercept of the tangent line at x0 is:
(y-f(x0))=f'(x0)*(x-x0) and solving for x:
x=x0-f(x0)/f'(x0)
x=x0-(x^x-5)/([ln(x)+1]*x^x)
simplifying:
x=x0-(1/[ln(x)+1])*(1-5/[x^x])
Since 2^2=4 and 3^3=27, with 2 as the closest integer, we'll use that as x0

x0=2
x1=2-(1/[0.693+1])*(1-5/4)=2.148
x2=2.148-(1/[0.764+1])*(1-5/5.164)=2.130
x3=2.130-(1/[0.756+1])*(1-5/5.003)=2.1293725
x4=x3-1/(ln(x3)+1)*(1-5/(x3^x3)=2.12937248276. So four iterations gives at least 12 decimal places (as many as the TI-89's numeric solver).

 By Albertfermat (Albertfermat) on Sunday, July 11, 2004 - 08:51 pm: Edit

i know for a fact that there is no "algebraic" method (besides approximations)

 By Tongos (Tongos) on Sunday, July 11, 2004 - 09:57 pm: Edit

check me on the cafe "nice ways to approximate", my method is very similar to that of newtons. its the "tongos method."

 By Stupid_Guy (Stupid_Guy) on Sunday, July 11, 2004 - 10:05 pm: Edit

SQRT(5) * SQRT(5) = 5

let our solution (SQRT 5) equals 2+X

(2+X) * (2+X) = 5
2^2 + 2*2*X + X^2 = 5
X^2 + 4X -1 = 0

now divide both sides of the equation by X^3

1/X + 4/X^2 - 1/X^3 = 0

now limit X --> Infinity

limX->inf 1/X + 4/X^2 - 1/X^3 = 0
so To solve the equation, X has to approach infinity.

now back to our original problem,

SQRT(5) = 2+X

= 2 + something that approaches infinity

= something that approaches infinity

solution:
SQRT(5) = something that approaches infinity

that's all folks

 By Tongos (Tongos) on Sunday, July 11, 2004 - 10:10 pm: Edit

 By Al0 (Al0) on Sunday, July 11, 2004 - 10:45 pm: Edit

That is because stupid guy's "solution" is stupid and abjectly wrong (I hope he meant it as a joke). He is solving for a quanitity q times itself (i.e. squared) and not raised to itself. Then through convulted and utterly flawed and pointless logic he proceeds to assert that the 2.236 (sqrt[5]) is equal to infinity. Again, I hope he was joking. But anyway,

Tongos -
how do you self-study math, I read as much as I can but usually end up with a mammoth list of prerequisites after too long. I usually discard the list and feel like I'm only scratching the surface. I'm asking you in particular since elsewhere you mentioned that usually learn math by proving all the theorems and, because this is what I do as well, I'd think whatever works for you would work for me as well. I've taken up through multi-variable, but just skip around through everything. Any tips on self-studying would be great, maybe we could even start a thread in the cafe for my edification. Future help greatly appreciated.

 By Tongos (Tongos) on Monday, July 12, 2004 - 12:12 am: Edit

i self study math daily. but i hide the formula under my fingers first. And i work diligently for as long as it takes to generate it. could be minutes or could be months. whatever you do, never make eye contact with the formula before you generate it, because it defeats the purpose. And make your work something that is satisfying to work on, don't be bored trying to generate stokes-navier equations or something, then you won't get anywhere.... pick something thats brain twisting to you. it's always great to have a challenge to work on.
The main problem when doing math without an instructor is finding a way to approach the problem. You know that there is a solution, you know that the given variables allows this, just how do you generate such. you have to look at all sides of the "algebra cube" first. see if it is possible to find it by stealthy SAT algebra. if not, you must make an effortful thought to arrive at the solution. "algebra does its thinking for you." it is so true.
Also, be ready for your approach. I know of many instances, in bed, when i find an awesome approach to the solution. the next morning, i start thinking, okay, now how do i begin the approach???
Lastly, keep an eye on what your doing. there are many instances when i do a math problem, trying for hours and hours. and by the last hour, i realize that i could've just generated the solution by using a simple equation or proof. and then i feel really dumb.
i agree, we should definatly start a thread in the cafe. but im going to shasta for six days or so, wont be back until saturday. so if you start a thread, please don't wonder where i am. (I know of several instances....no im kidding.)

 By Ultimatemath (Ultimatemath) on Monday, July 12, 2004 - 12:30 am: Edit

Where do you get books for self-studying? I wanna get into Chaos theory fractals. Or some other fun stuff

 By Tongos (Tongos) on Monday, July 12, 2004 - 12:33 am: Edit

wah, chaos theory fractals, wah, that's a bit of irony.

 By Tongos (Tongos) on Monday, July 12, 2004 - 12:36 am: Edit

amazons good, i dont know if thats what your looking for though. sleeptime.

 By Tongos (Tongos) on Saturday, July 17, 2004 - 09:09 pm: Edit

yeah, im back, lets start a thread.