自然對數 證明題

let f(x) = x -lnx
f(x)= ln(e^x) - ln(x)
=ln(e^x/x)
since (e^x) = lim(1+x/n)^n, n tends to + infinity
=(1+x+x^2/2!+x^3/3!+......)
therefore
f(x)=ln(e^x/x)=ln( 1+x+x^2/2!+......)/x
＞ln[(1+x)/x] ＞ln(1)=0
hence x-lnx＞０
x＞lnx
(這證法是從e的定義出發）

let f(x)=x-lnx
then f'(x)=1-1/x>0 for all x>1

and f'(x)<0 for all 0<1

so f(x) is min. when x=1

so f(x)>f(0)
x- lnx>1-ln1
x>1+lnx>lnx