\log(n) is what power of n?

David Troyer

David Troyer

Answered question

2022-01-01

log(n) is what power of n?

Answer & Explanation

temnimam2

temnimam2

Beginner2022-01-02Added 36 answers

No: If  is any positive number, then nϵ grows faster than logn. This can be clarified in the assertion.
limnlognnϵ=0 
for all ϵ>0. To prove this, just note that by LHospitals rule, 
limnlognnϵ=limn1nϵnϵ1=1ϵlimn1nϵ=0

einfachmoipf

einfachmoipf

Beginner2022-01-03Added 32 answers

Perhaps T. Bongers has answered the question you meant to ask, but given your mention of the definition of log I'm not so sure. To answer your question literally, the function nlog(n) is not equal to the function nnϵ for any number ϵ (not even if ϵ is "very small.) It is a different kind of function altogether, with very different properties, and it is certainly not defined as a power function.
Vasquez

Vasquez

Expert2022-01-09Added 669 answers

Suppose we seek such ϵ, that
nϵ=log(n).
Consider n>1. Then (if log(n) is natural logarithm, to the base e)
nϵ=eϵlog(n)log(n)=elog(log(n))then powers of e must be equal:ϵlog(n)=log(log(n)),ϵ=log(log(n))log(n).Examples:n=10:ϵ=0.3622156886...;n=102:ϵ=0.3316228421...;n=103:ϵ=0.2797789811...;n=106:ϵ=0.1900611565...;n=109:ϵ=0.1462731331....

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