Lecture 19 : Definition of the natural logarithmic function [Section 19.1]
(ii)
is differentiable and
for all .
(iii)
for all .
(iv)
for all and .
(v)
is strictly increasing and concave downward on . In particular, is a one-one function.
(vi)
as and as . In particular, it is an onto function and takes every real
value exactly once.
19.1.3
Definition:
By Theorem 19.1.2 (iv) above, there is a unique positive real number, denoted by , whose natural logarithm is equal to 1, i.e., . This positive real number is one of the Euler's number.
19.1.4
Note:
(i)
From Theorem 19.1.2 (iv), we have
hence, by the intermediate property and the fact, that is a strictly increasing, one-one function, we get . In fact one can show that .
(ii)
Recall that exists. Let it be denoted by . Since is continuous, we have
is differentiable and 
for all 
. 
for all 
. 
for all 
and 
. 
is strictly increasing and concave downward on 
. In particular, 
as 
and 
as 
. In particular, it is an onto function and takes every real 
, whose natural logarithm is equal to 1, i.e., 
. This positive real number 
is one of the Euler's number. 
is a strictly increasing, one-one function, we get 
. In fact one can show that 
. 
exists. Let it be denoted by 
. Since 
is continuous, we have 
