The bilinear transform is a first-order approximation of the natural logarithm function that is an exact mapping of the z-plane to the s-plane.

The logarithm of a complex number is thus a multi-valued function, because is multi-valued.

Neither the natural logarithm nor the square root functions can be continued analytically to an entire function.

This example assumes that the natural logarithm of a person's wage is a linear function of ( among other things ) the number of years of education that person has acquired.

The value distribution is similar to floating-point, but the value-to-representation curve, i. e. the graph of the logarithm function, is smooth ( except at 0 ).

The Bohr – Mollerup theorem states that among all functions extending the factorial functions to the positive real numbers, only the gamma function is log-convex, that is, its natural logarithm is convex on the positive real axis.

The principal branch of the complex logarithm function is holomorphic on the set C

* A binary symmetric channel ( BSC ) with crossover probability p is a binary input, binary output channel that flips the input bit with probability p. The BSC has a capacity of bits per channel use, where is the binary entropy function to the base 2 logarithm:

Consider the logarithm function: For any fixed base b, the logarithm function log < sub > b </ sub > maps from the positive real numbers R < sup >+</ sup > onto the real numbers R ; formally:

This mapping is one-to-one and onto, that is, it is a bijection from the domain to the codomain of the logarithm function.

In addition to being an isomorphism of sets, the logarithm function also preserves certain operations.

So the logarithm function is in fact a group isomorphism from the group ( R < sup >+</ sup >,< big >×</ big >) to the group ( R ,< big >+</ big >).

Graph of the natural logarithm function.

The natural logarithm function, if considered as a real-valued function of a real variable, is the inverse function of the exponential function, leading to the identities:

Thus, the logarithm function is an isomorphism from the group of positive real numbers under multiplication to the group of real numbers under addition, represented as a function:

) In fact, if this method is used, Newton inversion of the natural logarithm may conversely be used to calculate the exponential function efficiently.

This exponential function can be inverted to form a complex logarithm that exhibits most of the properties of the ordinary logarithm.

The logarithm must be taken to base e since the two terms following the logarithm are themselves base-e logarithms of expressions that are either factors of the density function or otherwise arise naturally.

For example, the present value at time 0 of a future payment at time t can be restated in the following way, where e is the base of the natural logarithm and r is the continuously compounded rate:

The following Python program computes the binary logarithm using the recursive method outlined above, showing the steps along the way:

In 17th century French the word's form, but not its meaning, changed to algorithm, following the model of the word logarithm, this form alluding to the ancient Greek arithmos = number.

* An equivalent formulation of the theorem is the following: if α and γ are nonzero algebraic numbers, and we take any non-zero logarithm of α, then ( log γ )/( log α ) is either rational or transcendental.

The formula also provides a negative slope, as can be seen from the following property of the logarithm:

The logarithm of G ( z + 1 ) has the following asymptotic expansion, as established by Barnes:

For a ND filter with optical density d the amount of optical power transmitted through the filter, which can be calculated from the logarithm of the ratio of the measurable intensity ( I ) after the filter to the incident intensity ( I < sub > 0 </ sub >), shown as the following:

In modern terms, prosthaphaeresis can be viewed as relying on the logarithm of complex numbers, in particular on the identity e ^( ix )= cos x + i sin x.

The converse is also true-any function which obeys the above relationship will be extensive.

Then it strictly obeys Kirchhoff's law of equality of radiative emissivity and absorptivity, with a black body source function.

A simple case ( sometimes called arithmetic quasiperiodic ) is if the function obeys the equation:

Another case ( sometimes called geometric quasiperiodic ) is if the function obeys the equation:

We have a language where is a constant symbol and is a unary function and the following axioms:

Area can be defined as a function from a collection M of special kind of plane figures ( termed measurable sets ) to the set of real numbers which satisfies the following properties:

The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T ′: X × Y → Z ′ is any bilinear function into a K-vector space Z ′, then only one linear function f: Z → Z ′ with exists.

/* The following function will print a non-negative number, n, to

On the other hand, for integer order α, the following relationship is valid ( note that the Gamma function becomes infinite for negative integer arguments ):

The auxiliary verb function derives from the copular function ; and, depending on one's point of view, one can still interpret the verb as a copula and the following verbal form as being adjectival.

It relies on the following equivalent definition of differentiability at a point: A function g is differentiable at a if there exists a real number g ′( a ) and a function ε ( h ) that tends to zero as h tends to zero, and furthermore

In 3 dimensions, a differential 0-form is simply a function f ( x, y, z ); a differential 1-form is the following expression: a differential 2-form is the formal sum: and a differential 3-form is defined by a single term: ( Here the a-coefficients are real functions ; the " wedge products ", e. g. can be interpreted as some kind of oriented area elements,, etc.

The function F is called universal if the following property holds: for every computable function f of a single variable there is a string w such that for all x, F ( w x ) = f ( x ); here w x represents the concatenation of the two strings w and x.

A real number a is said to be computable if it can be approximated by some computable function in the following manner: given any integer, the function produces an integer k such that:

Maximizing the log likelihood function with respect to x < sub > 0 </ sub > and γ produces the following system of equations:

For example, consider the following C ++ function:

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

It has the following universal property: if N is any complete metric space and f is any uniformly continuous function from M to N, then there exists a unique uniformly continuous function f ' from M ' to N, which extends f. The space M is determined up to isometry by this property, and is called the completion of M.

There is a product rule of the following type: if is a scalar valued function and F is a vector field, then

The following function returns the value of the “ if ” statement ( statements, too, produce results ), which evaluates to the value of either “ 1 ” or “ n * factorial ( n-1 )”:

The majority of patients go through a period of spontaneous recovery following brain injury in which they regain a great deal of language function.