Lecture 19 : Definition of the natural logarithmic function [Section 19.1]
19.1.2
Theorem:
Proof:
To prove (iv), we first note that for , by (iii), we have .
Hence, .
Also, .
Thus .
Hence, for every , .
Finally, if , then
To prove (v). Note that
Thus, is strictly increasing. Further,
Hence, is a concave function.
, by (iii), we have 
. 
. 
. 
. 
, 
. 
, then 
is strictly increasing. Further, 
is a concave function. 
