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Does Mathematica provide any way to compare two symbolic expressions and find which is the greatest? For example, I want to find which is the greatest of (n - 1) + 2(n - 1) Log[2, n] or n(n - 1)/2, where n is a variable.

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You mean Log2 (with a capital L), right? –  phantomas1234 Nov 24 '12 at 18:11
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1 Answer

Reduce[(n - 1) + 2 (n - 1) Log2[n] > n (n - 1)/2, n, Reals]

Returns that the statement is true for $$0 < n < \frac{-4}{\ln 2} \text{ProductLog}(\frac{-\ln 2}{4 \sqrt{2}})\lor 1 < n < \frac{-4}{\ln 2}\text{ProductLog}(\frac{-\ln 2}{4 \sqrt{2}})^*$$ Where $\text{ProductLog}$ is the solution to the Lambert W equation. As @Rojo notes in the comments, use N to find a numerical approximation.

General note: try using the free-form input in MMA to get a primary understanding for individual functions themselves and to learn of knew ones, such as Reduce.

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Wrap it in N to get 0. < n < 0.814262 || 1. < n < 18.9881 –  Rojo Nov 24 '12 at 18:25
    
A couple of pics: Plot[{(n - 1) + 2 (n - 1) Log2[n], n (n - 1)/2}, {n, -.2, 1.5}, GridLines -> {{0.814262, 1}, {}}]; Plot[{(n - 1) + 2 (n - 1) Log2[n], n (n - 1)/2}, {n, -.2, 20}, GridLines -> {{0.814262, 1, 18.9981}, {}}] –  David Carraher Nov 25 '12 at 1:22
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