exponential function defined as limit of powers

exponential function defined as limit of powers

It is possible to define the

exponential function

and the

natural logarithm

in terms of a limit of powers. In this entry, we shall present these
definitions after some background information and demonstrate the basic
properties of these

functions

from these definitions.

Two basic results which are needed to make this development possible
are the following:

Theorem 1

.

Let

x

be a

real number

and let

n

be an

integer

such that

n

>

0

and

n

+

x

>

0

. Then

(

n

+

x

n

)

n

<

(

n

+

1

+

x

n

+

1

)

n

+

1

.

Theorem 2

.

Suppose that

{

s

n

}

n

=

1

∞

is a

sequence

such that

lim

n

→

∞

⁡

n

⁢

s

n

=

0

. Then

lim

n

→

∞

⁡

(

1

+

s

n

)

n

=

1

,

For proofs, see the attachments. From them, we first conclude
that a sequence

converges

.

Theorem 3

.

Let

x

be any real number. Then the sequence

{

(

n

+

x

n

)

n

}

n

=

1

∞

is

convergent

.

The foregoing results show that the limit in the following
definition converges, and hence defines a bona fide function.

Definition 1

.

Let

x

be a real number. Then we define

exp

⁡

(

x

)

=

lim

n

→

∞

⁡

(

n

+

x

n

)

n

.

We may now derive some of the chief properties of this function.
starting with the

addition formula

.

Theorem 4

.

For any two real numbers

x

and

y

, we have

exp

⁡

(

x

+

y

)

=

exp

⁡

(

x

)

⁢

exp

⁡

(

y

)

.

Proof.

Since

n

⁢

(

n

+

x

+

y

)

(

n

+

x

)

⁢

(

n

+

y

)

=

1

-

x

⁢

y

(

n

+

x

)

⁢

(

n

+

y

)

and

lim

n

→

∞

⁡

n

(

n

+

x

)

⁢

(

n

+

y

)

=

0

,

theorem

2 above implies that

lim

n

→

∞

⁡

(

n

⁢

(

n

+

x

+

y

)

(

n

+

x

)

⁢

(

n

+

y

)

)

n

=

1

.

Since it permissible to multiply

convergent sequences

termwise, we have

exp

⁡

(

x

)

⁢

exp

⁡

(

y

)

=

lim

n

→

∞

⁡

(

n

+

x

n

)

n

⁢

(

n

+

y

n

)

n

⁢

(

n

⁢

(

n

+

x

+

y

)

(

n

+

x

)

⁢

(

n

+

y

)

)

n

=

lim

n

→

∞

⁡

(

n

+

x

+

y

n

)

n

=

exp

⁡

(

x

+

y

)

∎

Theorem 5

.

The function

exp

is

strictly increasing

.

Proof.

Suppose that

x

is a strictly positive real number. By theorem 1 and
the definition of the

exponential

as a limit, we have

1

+

x

<

exp

⁡

(

x

)

,
so we conclude that

0

<

x

implies

1

<

exp

⁡

(

x

)

.

Now, suppose that

x

and

y

are two real numbers with

x

>

y

. Since

x

-

y

>

0

, we have

exp

⁡

(

x

-

y

)

>

1

. Using theorem 4, we have

exp

⁡

(

x

)

=

exp

⁡

(

x

-

y

)

⁢

exp

⁡

(

y

)

>

exp

⁡

(

y

)

, so the function is strictly increasing.
∎

Theorem 6

.

The function

exp

is

continuous

.

Proof.

Suppose that

0

<

x

<

1

. By theorem 1 and the definition of the exponential
as a limit, we have

1

-

x

<

exp

⁡

(

-

x

)

and

1

+

x

<

exp

⁡

(

x

)

. By theorem 4,

exp

⁡

(

x

)

⁢

exp

⁡

(

-

x

)

=

exp

⁡

(

0

)

=

0

. Hence, we have the bounds

1

+

x

<

exp

⁡

(

x

)

<

1

/

(

1

-

x

)

and

1

-

x

<

exp

⁡

(

-

x

)

<

1

/

(

1

+

x

)

. From the former bound, we conclude
that

lim

x

→

0

-

⁡

exp

⁡

(

x

)

=

1

and, from the latter, that

lim

x

→

0

+

⁡

exp

⁡

(

x

)

=

1

, so

lim

x

→

0

⁡

exp

⁡

(

x

)

=

1

.

Suppose that

y

is any real number. By theorem 4,

exp

⁡

(

x

+

y

)

=

exp

⁡

(

x

)

⁢

exp

⁡

(

y

)

. Hence,

lim

x

→

0

⁡

exp

⁡

(

x

+

y

)

=

exp

⁡

(

y

)

⁢

lim

x

→

0

⁡

exp

⁡

(

x

)

=

exp

⁡

(

y

)

. In other words, for all real

y

, we have

lim

x

→

y

⁡

exp

⁡

(

x

)

=

exp

⁡

(

y

)

, so the exponential function is continuous.
∎

Theorem 7

.

The function

exp

is

one-to-one

and maps onto the positive real axis.

Proof.

The one-to-one property follows readily from

monotonicity

— if

exp

⁡

(

x

)

=

exp

⁡

(

y

)

, then we must have

x

=

y

, because otherwise, either

x

<

y

or

x

>

y

, which would imply

exp

⁡

(

x

)

<

exp

⁡

(

y

)

or

exp

⁡

(

x

)

>

exp

⁡

(

y

)

,
respectively. Next, suppose that

x

is a real number

greater

than

1

.
By theorem 1 and the definition of the exponential as a limit, we have

1

+

x

<

exp

⁡

(

x

)

. Thus,

1

<

x

<

exp

⁡

(

x

)

; since

exp

is continuous,
the

intermediate value theorem

asserts that there must exist a real
number

y

between

0

and

x

such that

exp

⁡

(

y

)

=

x

. If, instead,

0

<

x

<

1

, then

1

/

x

>

1

so we have a real number

y

such that

exp

⁡

(

y

)

=

1

/

x

. By theorem 4, we then have

exp

⁡

(

-

y

)

=

x

. So, given any real
number

x

>

0

, there exists a real number

y

such that

exp

⁡

(

y

)

=

x

,
hence the function maps onto the positive real axis.
∎

Theorem 8

.

The function

exp

is convex.

Proof.

Since the function is already known to be continuous, it suffices to show
that

exp

⁡

(

(

x

+

y

)

/

2

)

≤

(

exp

⁡

(

x

)

+

exp

⁡

(

y

)

)

/

2

for all real numbers

x

and

y

. Changing variables, this is

equivalent

to showing that

2

⁢

exp

⁡

(

a

+

b

)

≤

exp

⁡

(

a

)

+

exp

⁡

(

a

+

2

⁢

b

)

for all real numbers

a

and

b

.
By theorem 4, we have

exp

⁡

(

a

+

b

)

=

exp

⁡

(

a

)

⁢

exp

⁡

(

b

)

(1)

exp

⁡

(

a

+

2

⁢

b

)

=

exp

(

a

)

exp

(

b

)

2

.

(2)

Using the

inequality

2

⁢

x

≤

1

+

x

2

with

x

=

exp

⁡

(

b

)

and multiplying
by

exp

⁡

(

a

)

, we conclude that

2

⁢

exp

⁡

(

a

+

b

)

≤

exp

⁡

(

a

)

+

exp

⁡

(

a

+

2

⁢

b

)

,
hence the exponential function is convex.
∎

Defining the

constant

e

as

exp

⁡

(

1

)

, we find that the exponential
function gives powers of this number.

Theorem 9

.

For every real number

x

, we have

exp

⁡

(

x

)

=

e

x

.

Proof.

Applying an

induction

argument

to theorem 4, it can be shown that

exp

(

n

x

)

=

exp

(

x

)

n

for every real number

x

and every integer

n

. Hence, given a

rational number

m

/

n

, we have

exp

(

m

/

n

)

n

=

exp

(

m

)

=

exp

(

1

)

m

=

e

m

.
Thus,

e

⁢

x

⁢

p

⁢

(

m

/

n

)

=

e

m

/

n

so we see that

exp

⁡

(

x

)

=

e

x

when

x

is a
rational number. By

continuity

, it follows that

exp

⁡

(

x

)

=

e

x

for every
real number

x

.
∎

Title

exponential function defined as limit of powers

Canonical name

ExponentialFunctionDefinedAsLimitOfPowers

Date of creation

2013-03-22 17:01:35

Last modified on

2013-03-22 17:01:35

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

27

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 32A05

Related topic

ExponentialFunction

Related topic

ComplexExponentialFunction

Related topic

ExponentialFunctionNeverVanishes

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