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

$$provelnx⩽x$$

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

$$firstx>0letf\left(x\right)=x-ln\left(x\right)\Rightarrow$$$$f^{{}^{\prime }}\left(x\right)=1-\frac{1}{x}=\frac{x-1}{x}and{f}^{{}^{\prime }}\left(x\right)⩾0\iff x⩾1$$$${lim}_{x\to 0^{+}}f\left(x\right)=+\mathrm{\infty }and{lim}_{x\to +\mathrm{\infty }}f\left(x\right)=$$$${lim}_{x\to +\mathrm{\infty }}x(1-\frac{lnx}{x})={lim}_{+\mathrm{\infty }}x=+\mathrm{\infty }$$$$variationoff\left(x\right)$$$$x01+\mathrm{\infty }$$$$f^{{}^{\prime }}\left(x\right)-+$$$$f\left(x\right)+\mathrm{\infty }dec1inc+\mathrm{\infty }$$$$weseethatf\left(x\right)⩾0\mathrm{\forall }x>0\Rightarrow ln\left(x\right)⩽x\mathrm{\forall }x>0$$

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

$$thankssir$$

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

$$x<e^x$$$$\Rightarrow \ln x<x(not\ln x⩽x)$$

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