Which is greater: n 1.01 or n ⋅ l o g 10 ( n )?Can...

Jamison Rios

Jamison Rios

Answered

2022-07-15

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 < n l o g 10 ( n )
1000 1.01 < 1000 l o g 10 ( 1000 )
1071.51 < 3000

Answer & Explanation

Madden Luna

Madden Luna

Expert

2022-07-16Added 9 answers

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

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