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# Logarithms

The base **logarithm** is the inverse of the base exponential, i.e.

For example, , since . If the base is not specified, in mathematics it is assumed to be Euler's number since it is the base of the **natural logarithm**, although among scientists and engineers the tacit base might be 10 (**decimal logarithm**, **common logarithm**); they then use when the base is . The notation Leonhard Euler himself used was ,^{[1]} which thankfully has been changed to something a little clearer. In computer science, base 2 (**binary logarithm**) is often considered.

## Formulae

Since implies , we have

where denotes the **natural logarithm**.

## Maclaurin series expansions

Since for geometric series we have

thus

is the generating function of the harmonic sequence (unit fractions)

and (replacing by )

is the generating function of the alternating harmonic sequence

which sums to log(2), obtained by setting to 1, the convergence being assured by the alternating series test.

Also (since the fractions with even denominators cancel out)

is the generating function of the unit factions with odd denominators

and (since the fractions with odd denominators cancel out)

is the generating function of the unit factions with even denominators

## See also

- Arithmetic function templates:

#### Hierarchical list of operations pertaining to numbers ^{[2]} ^{[3]}

##### 0^{th} iteration

- Successor:

.S( *n*) - Predecessor:

.P( *n*)

##### 1^{st} iteration

- Addition:

, theS(S(⋯ " *a*times" ⋯ (S(*n*))))*sum*

, where*n*+*a*

is the*n**augend*and

is the*a**addend*. (When addition is commutative both are simply called*terms*.) - Subtraction:

, theP(P(⋯ " *s*times" ⋯ (P(*n*))))*difference*

, where*n*−*s*

is the*n**minuend*and

is the*s**subtrahend*.

##### 2^{nd} iteration

- Multiplication:

, the*n*+ (*n*+ (⋯ "*k*times" ⋯ (*n*+ (*n*))))*product*

, where*m*⋅*k*

is the*m**multiplicand*and

is the*k**multiplier*.^{[4]}(When multiplication is commutative both are simply called*factors*.) - Division: the
*ratio*

, where*n*/*d*

is the*n**dividend*and

is the*d**divisor*.- Quotient: (integer division).
- Remainder: (modulo and congruences).

##### 3^{rd} iteration

- Exponentiation (

as "degree",*d*

as "base",*b*

as "variable").*n*- Powers:

, written*n*⋅ (*n*⋅ (⋯ "*d*times" ⋯ (*n*⋅ (*n*))))

.*n**d* - Exponentials:

, written*b*⋅ (*b*⋅ (⋯ "*n*times" ⋯ (*b*⋅ (*b*))))

.*b**n*- Exponential function:

, where*e**n*

is Euler's number.*e*

- Exponential function:

- Powers:
- Exponentiation inverses (

as "degree",*d*

as "base",*b*

as "variable").*n*- Roots:

.*d*√*n* - Logarithms:

.log *b**n*- Natural logarithm function:

, orlog *n*

, wherelog *e**n*

is Euler's number.*e*

- Natural logarithm function:

- Roots:

##### 4^{th} iteration

- Tetration (

as "degree",*d*

as "base",*b*

as "variable").*n*- Tetra-powers (super-powers):

, written*n*^ (*n*^ (⋯ "*d*times" ⋯ (*n*^ (*n*))))

.*n*^^*d*or*n*↑↑*d* - Tetra-exponentials (super-exponentials):

, written*b*^ (*b*^ (⋯ "*n*times" ⋯ (*b*^ (*b*))))

.*b*^^*n*or*b*↑↑*n*

- Tetra-powers (super-powers):
- Tetration inverses (

as "degree",*d*

as "base",*b*

as "variable").*n*- Tetra-roots (super-roots)
- Tetra-logarithms (super-logarithms):

.slog *b**n*- Iterated logarithm:

.log ⁎ *b**n*= ⌈slog*b**n*⌉

- Iterated logarithm:

##### 5^{th} iteration

- Pentation (

as "degree",*d*

as "base",*b*

as "variable").*n*- Penta-powers:

, written*n*^^ (*n*^^ (⋯ "*d*times" ⋯ (*n*^^ (*n*^^ (*n*)))))

.*n*^^^*d*or*n*↑↑↑*d* - Penta-exponentials:

, written*b*^^ (*b*^^ (⋯ "*n*times" ⋯ (*b*^^ (*b*^^ (*b*)))))

.*b*^^^*n*or*b*↑↑↑*n*

- Penta-powers:
- Pentation inverses

##### 6^{th} iteration

- Hexation (

as "degree",*d*

as "base",*b*

as "variable").*n*- Hexa-powers:

, written*n*^^^ (*n*^^^ (⋯ "*d*times" ⋯ (*n*^^^ (*n*))))

.*n*^^^^*d*or*n*↑↑↑↑*d* - Hexa-exponentials:

, written*b*^^^ (*b*^^^ (⋯ "*n*times" ⋯ (*b*^^^ (*b*))))

.*b*^^^^*n*or*b*↑↑↑↑*n*

- Hexa-powers:
- Hexation inverses

##### 7^{th} iteration

- Heptation (

as "degree",*d*

as "base",*b*

as "variable").*n*- Hepta-powers:

, written*n*^^^^ (*n*^^^^ (⋯ "*d*times" ⋯ (*n*^^^^ (*n*))))

.*n*^^^^^*d*or*n*↑↑↑↑↑*d* - Hepta-exponentials:

, written*b*^^^^ (*b*^^^^ (⋯ "*n*times" ⋯ (*b*^^^^ (*b*))))

.*b*^^^^^*n*or*b*↑↑↑↑↑*n*

- Hepta-powers:
- Heptation inverses

##### 8^{th} iteration

- Octation (

as "degree",*d*

as "base",*b*

as "variable").*n*- Octa-powers:

, written*n*^^^^^ (*n*^^^^^ (⋯ "*d*times" ⋯ (*n*^^^^^ (*n*))))

.*n*^^^^^^*d*or*n*↑↑↑↑↑↑*d* - Octa-exponentials:

, written*b*^^^^^ (*b*^^^^^ (⋯ "*n*times" ⋯ (*b*^^^^^ (*b*))))

.*b*^^^^^^*n*or*b*↑↑↑↑↑↑*n*

- Octa-powers:
- Octation inverses

## Notes

- ↑ Ed Sandifer, "How Euler did it: Finding logarithms by hand," July 2005.
- ↑ Hyperoperation—Wikipedia.org.
- ↑ Grzegorczyk hierarchy—Wikipedia.org.
- ↑ There is a lack of consensus on which comes first. Having the multiplier come second makes it consistent with the definitions for exponentiation and higher operations. This is also the convention used with transfinite ordinals:

.*ω*× 2**:**=*ω*+*ω*