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 2009-11-04, 23:53 #1 Primeinator     "Kyle" Feb 2005 Somewhere near M50..sshh! 11011111102 Posts Exponential Inequality I came across a math puzzle on the Internet tonight and I was wondering what the method of solution would be. The inequality is as follows: ln x < x^.1 such that x is > 3 Note: Although there are an infinite number of solutions to this problem, the problem is asking for the smallest integer that satifies the equation. My guess was to use infinite series, though I'm not sure how to set it up. My math knowledge is limited to Differential Equations (Ordinary) and Calc 1-3. (Basic, not Advanced). Any insight would be greatly appreciated. Thanks Last fiddled with by Primeinator on 2009-11-04 at 23:54 Reason: Added the stipulation that x>3
2009-11-05, 03:30   #2
wblipp

"William"
May 2003
New Haven

2×32×131 Posts

Quote:
 Originally Posted by Primeinator Any insight would be greatly appreciated. Thanks
I'd start by letting z = ln(x) and solve for z.

 2009-11-05, 04:23 #3 axn     Jun 2003 13×192 Posts An N-R solution could be obtained, no? By Trial and error, looks like the solution lies between 3.4e15 and 3.5e15 Code: 3430631121407800 35.7715206395729718 35.7715206395729709 3430631121407801 35.7715206395729721 35.7715206395729719 3430631121407802 35.7715206395729724 35.7715206395729730 Last fiddled with by axn on 2009-11-05 at 04:34
2009-11-05, 05:42   #4
Primeinator

"Kyle"
Feb 2005
Somewhere near M50..sshh!

2×3×149 Posts

Quote:
 Originally Posted by axn An N-R solution could be obtained, no? By Trial and error, looks like the solution lies between 3.4e15 and 3.5e15 Code: 3430631121407800 35.7715206395729718 35.7715206395729709 3430631121407801 35.7715206395729721 35.7715206395729719 3430631121407802 35.7715206395729724 35.7715206395729730
I know the solution is an extremely large number (well, small compared to M47). Is there not a way to just work this out algebraically?

 2009-11-05, 06:54 #5 CRGreathouse     Aug 2006 31×191 Posts Pari gets the solution in < 10 milliseconds, even without transforming the problem. Code: solve(x=10,1e99,log(x)-x^.1)
2009-11-05, 06:55   #6
axn

Jun 2003

111258 Posts

Quote:
 Originally Posted by Primeinator I know the solution is an extremely large number (well, small compared to M47). Is there not a way to just work this out algebraically?
Heck, AFAIK, we can't even solve the general case of a deg-4 poly in one variable "algebraically" (or can we?).

N-R is your best bet. Though, since it is an integral solution we are searching for, we can do a simple "halving the interval" technique also. N-R requires that you have a reasonable starting position.

I proceeded by starting at x=3 and kept on doubling x until I got an upper bound. Then it's just a matter of applying whichever technique you want. Here, with a reasonable starting point, N-R will converge in < 5 iterations.

PS:- This particular problem is relatively easy to solve since both functions are monotonically increasing.

2009-11-05, 06:56   #7
axn

Jun 2003

125516 Posts

Quote:
 Originally Posted by CRGreathouse Pari gets the solution in < 10 milliseconds, even without transforming the problem. Code: solve(x=10,1e99,log(x)-x^.1)
Do you know PARI's algo/approach for solve?

2009-11-05, 07:03   #8
CRGreathouse

Aug 2006

31×191 Posts

Quote:
 Originally Posted by axn Do you know PARI's algo/approach for solve?
Brent's method, I think. Secant would be just as good in this case.

Newton-Raphson isn't fast unless you can compute derivatives quickly. The bisection method is slow, especially since you don't know a priori where to search.

Last fiddled with by CRGreathouse on 2009-11-05 at 07:05

2009-11-05, 08:34   #9

"Richard B. Woods"
Aug 2002
Wisconsin USA

1E0C16 Posts

Quote:
 Originally Posted by axn Heck, AFAIK, we can't even solve the general case of a deg-4 poly in one variable "algebraically" (or can we?).
http://mathworld.wolfram.com/QuarticEquation.html

2009-11-05, 10:14   #10
axn

Jun 2003

10010010101012 Posts

Quote:
Nice, but I don't think it quite does the trick. Apparently there is some cubic in between that you need to solve.

Of course, I only skimmed thru that page. If some one could post the canned solutions of a one-variable quartic, only in terms of its coeffs, much appreciated.

EDIT:- Or not. Apparently, cubic has closed form solution

Last fiddled with by axn on 2009-11-05 at 10:22

 2009-11-05, 13:13 #11 Orgasmic Troll Cranksta Rap Ayatollah     Jul 2003 641 Posts axn, you're thinking of degree 5 polynomials (and higher)

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