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# Logarithms

(Redirected from Logarithm)

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

$\log _{b}b^{y}:=y,\quad b>0,\,b\neq 1.\,$ For example, $\log _{7}2401=4$ , since $7^{4}=2401$ . If the base is not specified, in mathematics it is assumed to be Euler's number $e\,=\,2.71828\ldots \,$ 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 $\ln x$ when the base is $e$ . The notation Leonhard Euler himself used was $l\,x$ , which thankfully has been changed to something a little clearer. In computer science, base 2 (binary logarithm) is often considered.

## Formulae

Since $x=b^{y}$ implies $\log x=y\log b$ , we have

$y=\log _{b}x={\frac {\log x}{\log b}},\quad b>0,\,b\neq 1,\,$ where $\log$ denotes the natural logarithm.

## Maclaurin series expansions

Since for geometric series we have

${\frac {1}{1-x}}=\sum _{n=0}^{\infty }x^{n},\quad |x|<1,$ thus

$\log \left({\frac {1}{1-x}}\right)=-\log(1-x)=\int _{0}^{x}{\frac {du}{1-u}}=\int _{0}^{x}\sum _{n=0}^{\infty }u^{n}=\left.\sum _{n=0}^{\infty }{\frac {u^{n+1}}{n+1}}\right|_{0}^{x}=\sum _{n=0}^{\infty }{\frac {x^{n+1}}{n+1}}=\sum _{n=1}^{\infty }{\frac {x^{n}}{n}},\quad |x|<1,$ is the generating function of the harmonic sequence (unit fractions)

$\{{\tfrac {1}{1}},{\tfrac {1}{2}},{\tfrac {1}{3}},{\tfrac {1}{4}},{\tfrac {1}{5}},{\tfrac {1}{6}},{\tfrac {1}{7}},{\tfrac {1}{8}},{\tfrac {1}{9}},{\tfrac {1}{10}},{\tfrac {1}{11}},{\tfrac {1}{12}},\ldots \},$ and (replacing $x$ by $-x$ )

$\log(1+x)=\sum _{n=1}^{\infty }(-1)^{n+1}\,{\frac {x^{n}}{n}},\quad |x|<1,$ is the generating function of the alternating harmonic sequence

$\{{\tfrac {1}{1}},-{\tfrac {1}{2}},{\tfrac {1}{3}},-{\tfrac {1}{4}},{\tfrac {1}{5}},-{\tfrac {1}{6}},{\tfrac {1}{7}},-{\tfrac {1}{8}},{\tfrac {1}{9}},-{\tfrac {1}{10}},{\tfrac {1}{11}},-{\tfrac {1}{12}},\ldots \},$ which sums to log(2), obtained by setting $x$ to 1, the convergence being assured by the alternating series test.

Also (since the fractions with even denominators cancel out)

$\log {\sqrt {\frac {1+x}{1-x}}}={\frac {1}{2}}\log \left({\frac {1+x}{1-x}}\right)=\sum _{n=1}^{\infty }{\frac {x^{2n-1}}{2n-1}},\quad |x|<1,$ is the generating function of the unit factions with odd denominators

$\{{\tfrac {1}{1}},{\tfrac {1}{3}},{\tfrac {1}{5}},{\tfrac {1}{7}},{\tfrac {1}{9}},{\tfrac {1}{11}},{\tfrac {1}{13}},{\tfrac {1}{15}},{\tfrac {1}{17}},{\tfrac {1}{19}},{\tfrac {1}{21}},{\tfrac {1}{23}},\ldots \},$ and (since the fractions with odd denominators cancel out)

$\log {\sqrt {\frac {1}{1-x^{2}}}}={\frac {1}{2}}\log \left({\frac {1}{(1+x)(1-x)}}\right)=\sum _{n=1}^{\infty }{\frac {x^{2n}}{2n}},\quad |x|<1,$ is the generating function of the unit factions with even denominators

$\{{\tfrac {1}{2}},{\tfrac {1}{4}},{\tfrac {1}{6}},{\tfrac {1}{8}},{\tfrac {1}{10}},{\tfrac {1}{12}},{\tfrac {1}{14}},{\tfrac {1}{16}},{\tfrac {1}{18}},{\tfrac {1}{20}},{\tfrac {1}{22}},{\tfrac {1}{24}},\ldots \}.$ #### Hierarchical list of operations pertaining to numbers  

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