Logarithms

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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 $\scriptstyle e\,=\,2.71828\ldots \,$ 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

$\{{\tfrac {1}{2}},{\tfrac {1}{4}},{\tfrac {1}{6}},{\tfrac {1}{8}},{\tfrac {1}{10}},{\tfrac {1}{12}},{\tfrac {1}{14}},{\tfrac {1}{16}},{\tfrac {1}{18}},{\tfrac {1}{20}},{\tfrac {1}{22}},{\tfrac {1}{24}},\ldots \}.$

See also

• Natural logarithm function
• Iterated logarithm

• Arithmetic function templates:
  • {{log}} or {{log_e}} or {{ln}}
  • {{log_2}}
  • {{log_10}}
  • {{log_b}}
  • {{log-star}} ({{iterated logarithm}})

Hierarchical list of operations pertaining to numbers

0^th iteration

• Successor: 

  S(n)

  .
• Predecessor: 

  P(n)

  .

1^st 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.

2^nd 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.
  • Quotient: (integer division).
  • Remainder: (modulo and congruences).

3^rd 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

    .
    • Exponential function: 

      e n

      , where 

      e

      is Euler's number.
• Exponentiation inverses ( 

  d

  as "degree", 

  b

  as "base", 

  n

  as "variable").
  • Roots: 

    d √  n

    .
  • Logarithms: 

    logb n

    .
    • Natural logarithm function: 

      log n

      , or 

      loge n

      , where 

      e

      is Euler's number.

4^th iteration

• Tetration ( 

  d

  as "degree", 

  b

  as "base", 

  n

  as "variable").
  • Tetra-powers (super-powers): 

    n ^ (n ^ (⋯ "d times" ⋯ (n ^ (n))))

    , written 

    n ^^ d or n ↑↑ d

    .
  • Tetra-exponentials (super-exponentials): 

    b ^ (b ^ (⋯ "n times" ⋯ (b ^ (b))))

    , written 

    b ^^ n or b ↑↑ n

    .
• Tetration inverses ( 

  d

  as "degree", 

  b

  as "base", 

  n

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

    slogb n

    .
    • Iterated logarithm: 

      log ⁎b n = ⌈slogb n⌉

      .

5^th 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
  • Penta-roots
  • Penta-logarithms

6^th 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
  • Hexa-roots
  • Hexa-logarithms

7^th 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
  • Hepta-roots
  • Hepta-logarithms

8^th 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
  • Octa-roots
  • Octa-logarithms