Jamison Rios

2022-07-15

Which is greater: ${n}^{1.01}$ or $n\cdot lo{g}_{10}\left(n\right)$?
Can someone please explain how the right side can be less than the left side? I have plugged numerous numbers into n and every time I get the left side being less than the right side. My professor is convinced the right side is less than the left side. He has a PHD in math so he should be right. I just don't understand his explanation.
${n}^{1.01}
${1000}^{1.01}<1000\ast lo{g}_{10}\left(1000\right)$
$1071.51<3000$

Expert

In the race between a (positive) power and a logarithm, the power wins eventually. So ${n}^{0.01}>\mathrm{lg}\left(n\right)$ for all sufficiently large $n$, and thus ${n}^{1.01}>n\mathrm{lg}\left(n\right)$ for those same $n$. But how large is "sufficiently large"? In this case, $n>3.8125×{10}^{237}$ approximately.