natural logarithm

natural logarithm

The

natural logarithm

of a number is the

logarithm

in base of Euler’s number

e

. It can be defined as the map

ln

:

ℝ

+

→

ℝ

satisfying

ln

⁡

(

x

)

:=

∫

1

x

1

t

⁢

𝑑

t

.

(1)

Figure

1

shows the graph of

ln

.

Figure 1:

The graph of

ln

⁡

(

x

)

.

Instead of

ln

many mathematicians write

log

, physicists (and

calculators

) however consider

log

as the symbol for the logarithm in base 10.
One can show that the

function

defined in this way is the

inverse

of the

exponential function

. Indeed with equation (

1

) we have

d

d

⁢

x

⁢

ln

⁡

(

e

x

)

=

e

x

⋅

1

e

x

=

1

,

so there exists

C

∈

ℝ

such that

ln

⁡

(

e

x

)

=

x

+

C

.

Since

ln

⁡

(

e

0

)

=

0

we have

C

=

0

.
One can also prove that the above

integral

has the

defining properties

of a logarithm. For example if

x

,

y

∈

ℝ

+

we have

ln

⁡

(

x

⁢

y

)

=

∫

1

x

1

t

⁢

𝑑𝑡

+

∫

x

x

⁢

y

1

t

⁢

𝑑𝑡

=

ln

⁡

(

x

)

+

∫

x

x

⁢

y

1

t

⁢

𝑑

t

.

Now applying the substitution law with

u

:=

t

x

we have

∫

x

x

⁢

y

1

t

⁢

𝑑𝑡

=

∫

1

y

1

u

⁢

𝑑𝑢

=

ln

⁡

(

y

)

,

so we have

ln

⁡

(

x

⁢

y

)

=

ln

⁡

(

x

)

+

ln

⁡

(

y

)

.

The natural logarithm can also be represented as a

power series

around

1

. For

-

1

<

x

≤

1

we have

ln

⁡

(

1

+

x

)

=

∑

k

=

1

∞

(

-

1

)

k

+

1

k

⁢

x

k

.

(2)

This series is

divergent

at

x

=

-

1

but for

x

=

1

we have convergence due to Leibniz’s

theorem

and obtain

ln

⁡

(

2

)

=

∑

k

=

1

∞

(

-

1

)

k

+

1

k

.

In

real analysis

there is no reasonable way to extend the logarithm to

negative numbers

. In

complex analysis

the situation is a bit more complicated. Basically one can use the

Euler relation

to write a non-zero

complex number

z

in the form

z

=

R

⁢

e

i

⁢

φ

with

R

,

φ

∈

ℝ

.
We could try to define the complex logarithm

ln

⁡

(

z

)

to be

ln

⁡

(

R

)

+

i

⁢

φ

. However

φ

is unique only up to

addition

of a multiple of

2

⁢

π

.
While at first glance this does not appear to be very problematic, it actually prevents one from setting up a

continuous

logarithm on the

complex plane

(without 0, where the logarithm should be

infinite

). Say for example that we let the

imaginary part

of our logarithm take values from

-

π

to

π

. Then

lim

φ

↘

-

π

⁡

ln

⁡

e

i

⁢

φ

=

-

i

⁢

π

and

lim

φ

↗

π

⁡

ln

⁡

e

i

⁢

φ

=

i

⁢

π

while

e

i

⁢

φ

→

-

1

for both limits. Therefore the logarithm we defined is not continuous at -1. The same

argument

allows one to show that it is not continuous on the negative real numbers. In fact you can only define a continuous complex logarithm on a sliced plane, i.e. the complex plane with a half-line starting at 0 removed.

Title

natural logarithm

Canonical

name

NaturalLogarithm

Date of creation

2013-03-22 12:28:28

Last modified on

2013-03-22 12:28:28

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

13

Author

mathwizard (128)

Entry type

Definition

Classification

msc 33B10

Related topic

MatrixLogarithm

Related topic

ComplexLogarithm

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