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Simple inequality help

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I feel like an idiot: n^3>(2^n)/100 I need to solve this, but for the life of me I have no idea. There is a point of intersection around x=19, but I need an exact answer. Point in the right direction would be swell!

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It's not that simple.

A quick sketch of 2x and x3 should convince you that there are two roots to the inequality - one small and one larger, both positive. However, solution isn't possible in closed form. We can express the corresponding equality as a first-order exponential equation, and hence the solution can only be expressed in terms of the Lambert W function.

100x3 = 2x
ln(100x3) = ln(2x)
ln((d.x)3) = ln(2x) // d = 3√100
3ln(d.x) = xln(2)
x = A.ln(d.x) // A = 3/ln(2)

At this point, we convince ourselves that x is not isolatable. Is numerical solution viable? At least that should be fairly straightforward.

If you just need the answers, Maple tells me that in addition to two complex roots, the real solutions lie at:

0.2270471202
19.50008078

Admiral

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How did you get maple to output the 4 roots? It kept giving me only the first one, the very small one...

Anyways, thanks for the help!

//nm I got it. In maple the W(...) is the same as the link you posted?

[Edited by - _Sigma on May 24, 2007 5:38:37 PM]

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If you ask Maple (Version 9) to

solve(100*x^3 = 2^x);

it will give you a list of the four roots, in terms of the LambertW function, which is indeed the one I linked to. To get these values numerically, tell it to evaluate the result as floating values:

evalf(solve(100*x^3 = 2^x));

Admiral

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