prove-lnx-x-

Question Number 54730 by yorrick23ralph@gmail.com last updated on 09/Feb/19

p r o v e l n x ⩽ x

Commented by Abdo msup. last updated on 09/Feb/19

f i r s t x > 0 l e t f (x) = x - l n (x) \Rightarrow

f^{^{'}} (x) = 1 - \frac{1}{x} = \frac{x - 1}{x} a n d f^{^{'}} (x) ⩾ 0 \Leftrightarrow x ⩾ 1

{l i m}_{x \to 0^{+}} f (x) = + \infty a n d {l i m}_{x \to + \infty} f (x) =

{l i m}_{x \to + \infty} x (1 - \frac{l n x}{x}) = {l i m}_{+ \infty} x = + \infty

v a r i a t i o n o f f (x)

x 0 1 + \infty

f^{^{'}} (x) - +

f (x) + \infty d e c 1 i n c + \infty

w e s e e t h a t f (x) ⩾ 0 \forall x > 0 \Rightarrow l n (x) ⩽ x \forall x > 0

Commented by yorrick23ralph@gmail.com last updated on 09/Feb/19

t h a n k s s i r

Commented by mr W last updated on 10/Feb/19

x < e^{x}

\Rightarrow \ln x < x (n o t \ln x ⩽ x)

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