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How can I persuade Mathematica to help me find solutions to the inequality $k\ln(n) - k\ln(k) \leq u$ over the reals, assuming $u \geq n \geq k \geq 1$.

When I evaluate

Reduce[{k*Log[n] - k*Log[k] <= u, u > n, k > 1, n > k}, k, Reals]

Mathematica can't find a solution. However, $k=2, n=e^2, u = 8$ is a solution, for example.

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What sort of solution do you seek? FindInstance will find any (specified) number of solutions. – Michael E2 Mar 8 at 11:43
Would something like Reduce[k Log[Exp[2]] - k Log[k] <= 8, k, Reals] help ? – b.gatessucks Mar 8 at 11:47
Something in closed form. I don't mind if it has a number of cases. – Majid Mar 8 at 11:47
3  
RegionPlot[ k*Log[n] - k*Log[k] <= u && u > n && k > 1 && n > k /. u -> 8, {n, 0, 10}, {k, 0, 10}] – belisarius Mar 8 at 12:03
2  
@belisarius's triangle suggested I was misreading the formula. You can solve it by hand if you divide by k and know that x > Log[x] (or n/k > Log[n/k]). Forget FindInstance. – Michael E2 Mar 8 at 13:53
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