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I have trouble comparing the speed of growth of these two functions, could anyone help please?

$$\frac{10^{n-2}}{8^{n+3}} \;\;\;\text{and}\;\;\; ((\log n)^5)^n$$

Thank you in advance.

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Hint: compare the logs. – Did May 23 '11 at 11:06
Note that $10^{n-2} / 8^{n+3} = (1/51200) \times 1.25^n$ – Henry May 23 '11 at 11:37
The left-hand side is much less than $10^n$, and the right-hand side is much greater than $(\log n)^n$. Is it easier to compare these? – Jonas Meyer May 23 '11 at 12:02
The solution is simple and clear as ususal :) Thank you for your advice. – Ondrej Sotolar May 23 '11 at 12:15
Compare the $n$th root of these two functions. For the left one, it goes to $10/8$ because the difference between $n$, $n-2$, and $n+3$ becomes negligible when taking the roots, and for the right one, it goes to $\log(n)^5$. The latter is ultimately greater than $10/8$, so its $n$th power is also larger. – LuboŇ° Motl May 24 '11 at 7:04

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