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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]
0th iteration
1st iteration
- Addition:
S(S(⋯ "a times" ⋯ (S(n)))) |
, the sum , where is the augend and is the addend. (When addition is commutative both are simply called terms.)
- Subtraction:
P(P(⋯ "s times" ⋯ (P(n)))) |
, the difference , where is the minuend and is the subtrahend.
2nd iteration
- Multiplication:
n + (n + (⋯ "k times" ⋯ (n + (n)))) |
, the product , where is the multiplicand and is the multiplier.[4] (When multiplication is commutative both are simply called factors.)
- Division: the ratio , where is the dividend and is the divisor.
3rd iteration
- Exponentiation ( as "degree", as "base", as "variable").
- Powers:
n ⋅ (n ⋅ (⋯ "d times" ⋯ (n ⋅ (n)))) |
, written .
- Exponentials:
b ⋅ (b ⋅ (⋯ "n times" ⋯ (b ⋅ (b)))) |
, written .
- Exponentiation inverses ( as "degree", as "base", as "variable").
4th iteration
- Tetration ( as "degree", as "base", as "variable").
- Tetration inverses ( as "degree", as "base", as "variable").
5th iteration
- Pentation ( as "degree", as "base", as "variable").
- Pentation inverses
6th iteration
- Hexation ( as "degree", as "base", as "variable").
- Hexation inverses
7th iteration
- Heptation ( as "degree", as "base", as "variable").
- Heptation inverses
8th iteration
- Octation ( as "degree", as "base", as "variable").
- Octa-powers:
n ^^^^^ (n ^^^^^ (⋯ "d times" ⋯ (n ^^^^^ (n)))) |
, written .
- Octa-exponentials:
b ^^^^^ (b ^^^^^ (⋯ "n times" ⋯ (b ^^^^^ (b)))) |
, written .
- Octation inverses
Notes
Notes