Logarithms

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

${\log }_b{b}^y:=y,b>0,b\ne 1.$

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 $e=2.71828\dots$ 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 $lx$, 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

$y={\log }_{b}x=\frac{\log x}{\log b},b>0,b\ne 1,$

where $\log$ denotes the natural logarithm.

Maclaurin series expansions

Since for geometric series we have

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

thus

$\log \left(\frac{1}{1-x}\right)=-\log (1-x)=\underset{0}{\overset{x}{\int }}\frac{du}{1-u}=\underset{0}{\overset{x}{\int }}\sum _{n=0}^{\mathrm{\infty }}{u}^n={\sum _{n=0}^{\mathrm{\infty }}\frac{u^{n+1}}{n+1}|}_0^x=\sum _{n=0}^{\mathrm{\infty }}\frac{x^{n+1}}{n+1}=\sum _{n=1}^{\mathrm{\infty }}\frac{x^n}{n},|x|<1,$

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

$\{\frac{1}{1},\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6},\frac{1}{7},\frac{1}{8},\frac{1}{9},\frac{1}{10},\frac{1}{11},\frac{1}{12},\dots \},$

and (replacing $x$ by $-x$)

$\log (1+x)=\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n+1}\frac{x^n}{n},|x|<1,$

is the generating function of the alternating harmonic sequence

$\{\frac{1}{1},-\frac{1}{2},\frac{1}{3},-\frac{1}{4},\frac{1}{5},-\frac{1}{6},\frac{1}{7},-\frac{1}{8},\frac{1}{9},-\frac{1}{10},\frac{1}{11},-\frac{1}{12},\dots \},$

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)

$\log \sqrt{\frac{1+x}{1-x}}=\frac{1}{2}\log \left(\frac{1+x}{1-x}\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{x^{2n-1}}{2n-1},|x|<1,$

is the generating function of the unit factions with odd denominators

$\{\frac{1}{1},\frac{1}{3},\frac{1}{5},\frac{1}{7},\frac{1}{9},\frac{1}{11},\frac{1}{13},\frac{1}{15},\frac{1}{17},\frac{1}{19},\frac{1}{21},\frac{1}{23},\dots \},$

and (since the fractions with odd denominators cancel out)

$\log \sqrt{\frac{1}{1-x^2}}=\frac{1}{2}\log \left(\frac{1}{(1+x)(1-x)}\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{x^{2n}}{2n},|x|<1,$

is the generating function of the unit factions with even denominators

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

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