Log2

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed a

For fixed z

Values at fixed points

General characteristics

Domain and analyticity

(Subscript[log, a])(z) is an analytical function of a and z which is defined over (ℂ^2).

Symmetries and periodicities

Mirror symmetry

Quasi-permutation symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

For fixed a, the function (Subscript[log, a])(z) does not have poles and essential singularities.

With respect to a

For fixed z, the function (Subscript[log, a])(z)has one simple pole at point a= 1with residue log(z).

Branch points

With respect to z

With respect to a

Branch cuts

With respect to z

With respect to a

Analytic continuations

(Subscript[log, a])(z) is chosen to be the principal branch of the general logarithmic function (Subscript[(OverTilde[log]), a])(z) which has double infinitely many sheets: (Subscript[(OverTilde[log]), a])(z)=(log(z)+2ℼ i k)/(log(a)+2ℼ i l), k,l∈ℤ.

Series representations

Generalized power series

Expansions at a==1

Expansions at z==1

Expansions of Log[a,1+z] at z==0

Asymptotic series expansions

Differential equations

Ordinary linear differential equations and wronskians

With respect to z

Ordinary nonlinear differential equations

With respect to a

Partial differential equations

Transformations

Multiple arguments

Power of arguments

Products, sums, and powers of the direct function

Products of the direct function

Sums of the direct function

Identities

Functional identities

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Differentiation

Low-order differentiation

With respect to a

With respect to z

Symbolic differentiation

With respect to a

With respect to z

Fractional integro-differentiation

With respect to a

With respect to z

Integration

Indefinite integration

Involving only one direct function

Involving only one direct function with respect to a

Involving one direct function and elementary functions with respect to a

Representations through more general functions

Through Meijer G

Classical cases for the direct function itself

Representations through equivalent functions

With inverse function

The left side of above formula corresponds to composition f((f^((-1)))(z))/; f(z)==(Subscript[log, a])(z), which generically does not equal to z.

The left side of above formula corresponds to composition f((f^((-1)))(z))/; f(z)==(Subscript[log, a])(z), which equal to z under restriction -π<Im(z log(a))≤π.

The left side of above formula corresponds to composition (f^((-1)))(f(z))/; f(z)==(Subscript[log, a])(z), which generically equal to z.

The left side of above formula corresponds to composition f((f^((-1)))(a))/; f(a)==(Subscript[log, a])(z), which generically does not equal to a.

The left side of above formula corresponds to composition f((f^((-1)))(a))/; f(a)==(Subscript[log, a])(z),  which equal to a under restriction -π<Im(((log(z))/a))≤π.

The left side of above formula corresponds to composition (f^((-1)))(f(a))/; f(a)==(Subscript[log, a])(z), which generically equal to a.

With related functions

Zeros

Theorems

Benford's law

The probability (Subscript[p, n]) that the first digit in a large set of data is n (n=1,2,…,9) is given by  (Subscript[p, n])==(Subscript[log, 10])(((n+1)/n)).

The probability for the number n to occur in the normal continued fraction expansion

For almost all real numbers x , the probability (Subscript[p, n])of occurrence of the integer n in the normal continued fraction expansion is given by (Subscript[p, n])==-(Subscript[log, 2])(1-(1/(((n+1))^2))).

History

– J. Napier (1614) published the first tables and used word Log
– H. Briggs (1617) published the first tables in base 10 and found logarithms of the first 25 primes
– J. Burgi (1620)
– J. Kepler (1624)
– B. Cavalieri (1632)
– J. Gregory (1668) found series expansion for log
– N. Mercator (1668) used "Log naturalis"
– J. N. Lambert (1770) and J.-L. Lagrange (1776) found continued fraction representations for Log
– L. Euler (1749) found that log was multivalued

The function log is encountered often in mathematics and the natural sciences.

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