Thus e x e x 1 x e x 1 x 1 x e x therefore de x d x e

Thus
e
x
(e
∆ x
− 1)
∆ x
=
e
x
(1 + ∆ x − 1)
∆ x
= e
x
.
Therefore
de
x
d x
= e
x
. (1.11)
The function e x (or written as exp( x )) is generally called the natural exponen-
tial function, or simply the exponential function. Not only is the exponential
function equal to its own derivative, it is the only function (apart from a mul-
tiplication constant) that has this property. Because of this, the exponential
function plays a central role in mathematics and sciences.

1.3 A Peculiar Number Called e 19
1.3.2 The Natural Logarithm
If e
y
= x, then by definition
y = log
e
x.
The logarithm to the base e is known as the natural logarithm. It appears
with amazing frequency in mathematics and its applications. So we give it a
special symbol. It is written as ln x . That is
y = log
e
x = ln x.
Thus
e
ln x
= x.
Furthermore,
ln e
x
= x ln e = x.
In this sense, the exponential function and the natural logarithm are inverses
of each other.
Example 1.3.1. Find the value of ln 10.

20 1 Complex Numbers
for a very small t . Since ∆ x approaches zero as a limit, for any fixed x ,
∆ x
x
can certainly be made as small as we wish. Therefore, we can set
∆ x
x
= t , and
conclude
∆ x
x
= ln 1 +
∆ x
x
.
Thus,
d(ln x )
d x
= lim
∆ x → 0
1
∆ x
ln 1 +
∆ x
x
= lim
∆ x → 0
1
∆ x
∆ x
x
=
1
x
.
This in turn means
d (ln x ) =
d x
x
or
d x
x
= ln x + c, (1.12)
where c is the constant of integration. It is well known that because of
d x
n +1
d x
= ( n + 1) x
n
,
we have
x
n
d x =
x n +1
( n + 1)
+ c.
This formula holds for all values of n except for n = − 1, since then the
denominator n + 1 is zero. This had been a difficult problem, but now we see
that (1.12) provides the “missing case.”
In numerous phenomena, ranging from population growth to the decay of
radioactive material, in which the rate of change of some quantity is propor-
tional to the quantity itself. Such phenomenon is governed by the differential
equation
d y
d t
= ky,
where k is a constant that is positive if y is increasing and negative if y is
decreasing. To solve this equation, we write it as
d y
y
= k d t
and then integrate both sides to get
ln y = kt + c, or y = e
kt + c
= e
kt
e
c
.
If y
0
denotes the value of y when t = 0, then y
0
= e
c
and
y = y
0
e
kt
.
This equation is called the law of exponential change.

1.4 The Exponential Function as an Infinite Series 21
1.3.3 Approximate Value of e
The number e is found of such great importance, but what is the numerical
value of e, which we have, so far, defined as 10
(1 / 2 . 3025)
? We can use our table
of successive square root of 10 to calculate this number. The powers of 10
are given in the first column of Table 1.1. If we can find a series of numbers
n
1
, n
2
, n
3
, . . . in this column, such that
1
2 . 3026
= n
1
+ n
2
+ n
3
+ · · · ,
then
10
1
2 . 3026
= 10
n
1
+ n
2
+ n
3
+ ···
= 10
n
1
10
n
2
10
n
3
· · · .
We can read from the second column of the table 10
n
1
, and 10
n
2
, and 10
n
3
and so on, and multiply them together. Let us do just that.