Which is greater: n 1.01 or n ⋅ l o g 10 ( n )?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 &lt; n ⋅ l o g 10 ( n ) 1000 1.01 &lt; 1000 ∗ l o g 10 ( 1000 ) 1071.51 &lt; 3000

Question

Which is greater:       n          1.01       or   n  ⋅  l  o      g          10        (  n  )?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&#039;t understand his explanation.      n          1.01        &amp;lt;  n  ⋅  l  o      g          10        (  n  )      1000          1.01        &amp;lt;  1000  ∗  l  o      g          10        (  1000  )  1071.51  &amp;lt;  3000

Madden Luna · Accepted Answer

In the race between a (positive) power and a logarithm, the power wins eventually. So       n          0.01        &amp;gt;  lg  ⁡  (  n  ) for all sufficiently large   n, and thus       n          1.01        &amp;gt;  n  lg  ⁡  (  n  ) for those same   n. But how large is &quot;sufficiently large&quot;? In this case,   n  &amp;gt;  3.8125  ×      10          237       approximately.