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Old 2009-11-04, 23:53   #1
Primeinator
 
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"Kyle"
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Default 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
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Old 2009-11-05, 03:30   #2
wblipp
 
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Quote:
Originally Posted by Primeinator View Post
Any insight would be greatly appreciated. Thanks
I'd start by letting z = ln(x) and solve for z.
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Old 2009-11-05, 04:23   #3
axn
 
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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
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Old 2009-11-05, 05:42   #4
Primeinator
 
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Quote:
Originally Posted by axn View Post
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?
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Old 2009-11-05, 06:54   #5
CRGreathouse
 
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Pari gets the solution in < 10 milliseconds, even without transforming the problem.
Code:
solve(x=10,1e99,log(x)-x^.1)
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Old 2009-11-05, 06:55   #6
axn
 
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Quote:
Originally Posted by Primeinator View Post
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.
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Old 2009-11-05, 06:56   #7
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Quote:
Originally Posted by CRGreathouse View Post
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?
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Old 2009-11-05, 07:03   #8
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Quote:
Originally Posted by axn View Post
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
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Old 2009-11-05, 08:34   #9
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Quote:
Originally Posted by axn View Post
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
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Old 2009-11-05, 10:14   #10
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Quote:
Originally Posted by cheesehead View Post
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
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Old 2009-11-05, 13:13   #11
Orgasmic Troll
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axn, you're thinking of degree 5 polynomials (and higher)
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