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If one algorithm has a running time of $100n^2$ and another of $2^n$; how can I find the smallest value of $n$ such that the former is faster than the latter?

I could do: $100n^2 < 2^n$ then $\ln(100n^2) < n\ln(2)$ but how do I simplify the left side?

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    $\begingroup$ You try a few values of $n$, get some feeling for what's going on, try a few more values of $n$, and zero in on the answer. $\endgroup$
    – Gerry Myerson
    Jul 5, 2013 at 12:44
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    $\begingroup$ An explicit form for the intersections of the two graphs is difficult to obtain. An iterative/heuristic approach probably your best shot: W|A link $\endgroup$
    – apnorton
    Jul 6, 2013 at 3:23

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The best method could be to plot a rough graph of 100*n^2 and 2^n on the same pair of axes. Then see a probable point of intersection by some iterative method.

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