Lecture 19 : Definition of the natural logarithmic function [Section 19.1]
19.1.2
Theorem:
Proof:
It is obvious from the definition that . Further, being the integral of a positive function, for .
For , .
This proves (i).
Since is a continuous function, by FCT-II, is differentiable and
for all ,
proving (ii). To prove (iii), fix arbitrarily. Let and for every .
Then for every .
Thus, for every and a constant .
Since, , we have . Hence, for all .
This proves (iii).
. Further, 
being the integral of a positive function, 
for 
. 
, 
. 
is a continuous function, by FCT-II, 
is differentiable and 
for all 
, 
arbitrarily. Let 
and 
for every 
. 
for every 
. 
for every 
and a constant 
. 
, we have 
. Hence, 
for all 
. 
