rigorous definition of the logarithm

rigorous definition of the logarithm

In this entry, we shall construct the logarithm as a

Dedekind cut

and then demonstrate some of its basic properties. All that is
required in the way of background material are the properties of

integer

powers of

real numbers

.

Theorem 1

.

Suppose that

a

,

b

,

c

,

d

are

positive integers

such that

a

/

b

=

c

/

d

and that

x

>

0

and

y

>

0

are real numbers.
Then

x

a

≤

y

b

if and only if

x

c

≤

y

d

.

Proof.

Cross multiplying, the condition

a

/

b

=

c

/

d

is

equivalent

to

a

⁢

d

=

b

⁢

c

.
By

elementary

properties of powers,

x

a

≤

y

b

if and only if

x

a

⁢

d

≤

y

b

⁢

d

. Likewise,

x

c

≤

x

d

if and only if

x

b

⁢

c

≤

y

b

⁢

d

which, since

b

⁢

c

=

a

⁢

d

, is equivalent to

x

a

⁢

d

≤

y

b

⁢

d

.
Hence,

x

a

≤

y

b

if and only if

x

c

≤

x

d

.
∎

Theorem 2

.

Suppose that

a

,

b

,

c

,

d

are positive integers such that

a

/

b

≤

c

/

d

and that

x

>

1

and

y

>

0

are real numbers.
If

x

c

≤

y

d

then

x

a

≤

y

b

.

Proof.

Since we assumed that

b

>

0

, we have that

x

c

≤

y

d

is equivalent
to

x

b

⁢

c

≤

y

b

⁢

d

. Likewise, since

d

>

0

, we have that

x

a

≤

y

b

is equivalent to

x

a

⁢

d

≤

y

b

⁢

d

. Cross-multiplying,

a

/

b

≤

c

/

d

is equivalent to

a

⁢

d

≤

b

⁢

c

. Since

x

>

1

, we have

x

a

⁢

d

≤

x

b

⁢

c

.
Combining the above statements, we conclude that

x

c

≤

y

d

implies

x

a

≤

y

b

.
∎

Theorem 3

.

Suppose that

a

,

b

,

c

,

d

are positive integers such that

a

/

b

>

c

/

d

and that

x

>

1

and

y

>

0

are real numbers.
If

x

a

>

y

b

then

x

c

>

y

d

.

Proof.

Since we assumed that

b

>

0

, we have that

x

c

>

y

d

is equivalent
to

x

b

⁢

c

>

y

b

⁢

d

. Likewise, since

d

>

0

, we have that

x

a

>

y

b

is equivalent to

x

a

⁢

d

>

y

b

⁢

d

. Cross-multiplying,

a

/

b

>

c

/

d

is equivalent to

a

⁢

d

>

b

⁢

c

. Since

x

>

1

, we have

x

a

⁢

d

>

x

b

⁢

c

.
Combining the above statements, we conclude that

x

c

>

y

d

implies

x

a

>

y

b

.
∎

Theorem 4

.

Let

x

>

1

and

y

be real numbers.
Then there exists an integer

n

such that

x

n

>

y

.

Proof.

Write

x

=

1

+

h

. Then we have

(

1

+

h

)

n

≥

1

+

n

⁢

h

for all
positive integers

n

. This fact is easily proved by

induction

.
When

n

=

1

, it reduces to the triviality

1

+

h

≥

h

. If

(

1

+

h

)

n

≥

1

+

n

⁢

h

, then

(

1

+

h

)

n

+

1

=

(

1

+

h

)

⁢

(

1

+

h

)

n

≥

(

1

+

h

)

⁢

(

1

+

n

⁢

h

)

=

1

+

(

n

+

1

)

⁢

h

+

n

⁢

h

2

≥

1

+

(

n

+

1

)

⁢

h

.

By the

Archimedean property

, there exists an integer

n

such
that

1

+

n

⁢

h

>

y

, so

x

n

>

y

.
∎

Theorem 5

.

Let

x

>

1

and

y

be real numbers.
Then the pair of sets

(

L

,

U

)

where

L

=

{

r

∈

ℚ

∣

(

∃

a

,

b

∈

ℤ

)

b

>

0

∧

r

=

a

/

b

∧

x

a

≤

y

b

}

(1)

U

=

{

r

∈

ℚ

∣

(

∃

a

,

b

∈

ℤ

)

b

>

0

∧

r

=

a

/

b

∧

x

a

>

y

b

}

(2)

forms a Dedekind cut.

Proof.

Let

r

be any

rational number

. Then we have

r

=

a

/

b

for some integers

a

and

b

such that

b

>

0

. The possibilities

x

a

≤

y

b

and

x

a

>

y

b

are exhaustive so

r

must belong to at least one of

U

and

L

. By

theorem

1, it cannot belong to both. By theorem 2, if

r

∈

L

and

s

≤

r

, then

s

∈

L

as well. By theorem 3, if

r

∈

U

and

s

>

r

, then

s

∈

U

as well. By theorem 4, neither

L

nor

U

are empty. Hence,

(

L

,

U

)

is a Dedekind cut and defines a
real number.
∎

Definition 1

.

Suppose

x

>

1

and

y

>

0

are real numbers. Then,
we define

log

x

⁡

y

to be the real number defined by the cut

(

L

,

U

)

of
the above theorem.

Title

rigorous

definition of the logarithm

Canonical name

RigorousDefinitionOfTheLogarithm

Date of creation

2013-03-22 17:00:37

Last modified on

2013-03-22 17:00:37

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

18

Author

rspuzio (6075)

Entry type

Derivation

Classification

msc 26A06

Classification

msc 26A09

Classification

msc 26-00

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