WOLFRAM|DEMONSTRATIONS PROJECT

The Natural Logarithm is the Limit of the Integrals of Powers

​
power a
1
Assume that
a≠0
and that
x>0
.
The integral of
a-1
x
is
a
x
a
+C
, where
C
is an arbitrary constant. The integral of
-1
x
=
1
x
is
log(x)+C'
, where again
C'
is an arbitrary constant and
log(x)
is the natural logarithm of
x
, often written as
ln(x)
.
When
a
is close to zero,
a-1
x
and
-1
x
are close, so there must be some connection between their integrals!
Choose
C=-
1
a
and
C'=0
so that the two integrals are both zero at
x=1
. The integrals are then
a
x
a
-
1
a
and
log(x)
. For
a
close to zero these functions are very close; in symbols,
lim
a0
a
x
a
-
1
a
=log(x)
.
Using the difference quotient for the derivative of the base-
x
exponential function
f(b)=
b
x
with respect to
b
(not
x
) and using
a
instead of the more usual
h
gives
f'(b)=
lim
a0
f(b+a)-f(b)
a
=
lim
a0
b+a
x
-
b
x
a
=
b
x
lim
a0
a
x
-1
a
=
b
x
log(x)
. This is more usually written with
x
as the variable:
d
dx
x
u
=
x
u
log(u)
, with the special case
d
dx
x
e
=
x
e
.