How to show log a<= n(root[n](a)-1) <= sqrt[n](a) log a

Nelson Jennings 2022-07-16 Answered
How to show log a n ( a n 1 ) a n log a
Let b = a n
How to show: log a n ( a n 1 ) a n log a?
Thank you ;)
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Answers (1)

tiltat9h
Answered 2022-07-17 Author has 14 answers
Noting a = b n , this is equivalent to
n log b n ( b 1 ) n b log b ,
i.e. just
To get this, employ the mean value theorem separately for b 1 and 0 < b < 1 (I assume the condition b > 0 as log isn't defined for non-positive numbers). Notice first it does work for b = 1, as all 3 quantities are 0. If b > 1, applying the mean value theorem to x log x gives us that there exists c [ 1 , b ] such that
log b log 1 = 1 c ( b 1 ) ,
then noting log 1 = 0, and using the facts c [ 1 , b ] and b 1 are positive gives us
1 b ( b 1 ) 1 c ( b 1 ) b 1
gives us the result we want after a little bit of re-arranging. For b < 1, the working is pretty much the same, except we get the last inequality from the fact c [ b , 1 ] and b 1 is negative instead.

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