Lecture 19 : Definition of the natural logarithmic function [Section 19.1]
19.1
Definition of the natural logarithmic function :
Recall, that in elementary calculus, one uses the formula ,
without really knowing the function . We can use this formula, and the fundamental theorem of calculus, to define the . Since the function is continuous on , it is integrable, i.e., exists for every . This motivate our definition:
19.1.1
Definition:
For , define the natural logarithm of by .
We show that the above function is the familiar logarithmic function.
, 
. We can use this formula, and the fundamental theorem of calculus, to define the 
. Since the function 
is continuous on 
, it is integrable, i.e., 
exists for every 
. This motivate our definition: 
, define the 
by 
. 
has the following properties: 
.
for 
and 
for 
. 
