What is the minimum degree of x so that it is greater than or equal to ln(x)? I was thinking of this question and couldn't find it anywhere. I was trying to find a solution by finding the maximum of the function f(x) = (ln(ln(x)))/(ln(x)) yet I'm not sure if that's gonna work. Thanks in advance.

TunaIeu

TunaIeu

Answered question

2022-11-22

What is the minimum degree of x so that it is greater than or equal to ln(x)?
I was thinking of this question and couldn't find it anywhere. I was trying to find a solution by finding the maximum of the function f ( x ) = l n ( l n ( x ) ) l n ( x )
yet I'm not sure if that's gonna work. Thanks in advance.

Answer & Explanation

Desirae Wu

Desirae Wu

Beginner2022-11-23Added 10 answers

We can solve this by using calculus:
Let
f ( x ) = ln ( ln ( x ) ) ln ( x )
Then, taking the derivative:
f ( x ) = 1 ln ( ln ( x ) ) x ln 2 ( x )
Setting it equal to 0 to find the maximum value:
1 ln ( ln ( x ) ) x ln 2 ( x ) = 0
1 ln ( ln ( x ) ) = 0
ln ( x ) = e
x = e e
Our maximum value occurs when x = e e Plugging this back into our function:
f ( e e ) = ln ( ln ( e e ) ) ln ( e e )
f ( e e ) = 1 e

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