Lower bound for logarithm? Given a real c such that 1 < c , is there any known and d

Trevor Wood

Trevor Wood

Answered question

2022-05-28

Lower bound for logarithm?
Given a real c such that 1 < c, is there any known and direct lower bound, other than 0, for ( ln c ), i.e., A < ln c
Thanks

Answer & Explanation

exhumatql

exhumatql

Beginner2022-05-29Added 12 answers

For all x 1 we have log x x 1 x + 1 , with equality only at x = 1,
Note that the derivative of log x is 1 / x, and the derivative of the rational function, which we can rewrite as 1 2 ( x + 1 ) 1 , is 2 ( x + 1 ) 2 :
x 1 1 x > 2 ( x + 1 ) 2
Antoine Hill

Antoine Hill

Beginner2022-05-30Added 4 answers

Since lim x 1 ln x = 0 , there can be no positive lower bound for the natural logarithms of numbers greater than 1
More explicitly, given A > 0 you can always find c > 1 such that 0 < ln c < A: just take c = e A / 2 .

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