In mathematics, many logarithmic identities exist. The following is a compilation of the notable of these, many of which are used for computational purposes.

Trivial identities

Trivial mathematical identities are relatively simple (for an experienced mathematician), though not necessarily unimportant. The trivial logarithmic identities are as follows:

Explanations

By definition, we know that:

.

, we can see that:

So, substituting these values into the formula, we see that:

which gets us the first property.

, we can see that:

So, substituting these values into the formula, we see that:

which gets us the second property.

Cancelling exponentials

Logarithms and exponentials with the same base cancel each other. This is true because logarithms and exponentials are inverse operationsmuch like the same way multiplication and division are inverse operations, and addition and subtraction are inverse operations:

may not be referring to the same number)

, we get the following equation:

which gets us the first equation.

.

, we get the following equation:

which gets us the second equation.

.

Using simpler operations

Logarithms can be used to make calculations easier. For example, two numbers can be multiplied just by using a logarithm table and adding. These are often known as logarithmic properties, which are documented in the table below. The first three operations below assume that and/or , so that and . Derivations also use the log definitions and . are positive real numbers are real numbers.

The laws result from canceling exponentials and the appropriate law of indices. Starting with the first law:

The law for powers exploits another of the laws of indices:

The law relating to quotients then follows:

Similarly, the root law is derived by rewriting the root as a reciprocal power:

Derivations of product, quotient, and power rules

These are the three main logarithm laws, rules, or principles, from which the other properties listed above can be proven. Each of these logarithm properties correspond to their respective exponent law, and their derivations and proofs will hinge on those facts. There are multiple ways to derive or prove each logarithm lawthis is just one possible method.

Logarithm of a product

To state the logarithm of a product law formally:

Derivation:

.

Rewriting these as exponentials, we see that

) using exponent laws as

to both sides of the equality.

, giving

.

This completes the derivation.

Logarithm of a quotient

To state the logarithm of a quotient law formally:

Derivation:

.

.

Rewriting these as exponentials, we see that:

) using exponent laws as

to both sides of the equality.

, giving

.

This completes the derivation.

Logarithm of a power

To state the logarithm of a power law formally:

Derivation:

.

, so we have

:

.

to both sides of the equality.

.

, rearranging, and simplifying gives

This completes the derivation.

Changing the base

Most calculators have buttons for natural logarithms () and common logarithms (log or ), but not all calculators have buttons for the logarithm of an arbitrary base. Accordingly, it is sometimes useful to change the base of a logarithm.As briefly mentioned in the Solving for in the exponential equation For example, the binary logarithm (), which is widely used in

Proof and derivation

are the two bases we will be using for the logarithms. They cannot be 1, because the logarithm function is not well defined for the base of 1..

to both sides of the equality,

,

, we get the following:

back into the equation,

.

This formula has several consequences:

is any permutation of the subscripts . For example

Summation and subtraction

The following summation and subtraction rule is especially useful in probability theory when one is dealing with a sum of log-probabilities: to avoid losing the "1 +" due to rounding errors. Many programming languages have a specific log1p(x) is small).

More generally:

Exponents

A useful identity involving exponents:

or more universally:

Other or resulting identities

Inequalities

Based on,

, but not for large numbers.

Tropical identities

The following identity relates log semiring to the min-plus semiring.

Calculus identities

Limits

The last limit is often summarized as "logarithms grow more slowly than any power or root of x".

Derivatives of logarithmic functions

Integral definition

, we change the order of integration, which changes the sign of the integral:

Therefore:

Riemann Sum

is a sample point in each interval.

Series representation

has a well-known Taylor series in the open-closed interval :

, the series is conditionally convergent, and for all other values, it is absolutely convergent. In these cases, different representations or methods must be used to evaluate the logarithm.

Harmonic number difference

It is not uncommon in advanced mathematics, particularly in analytic number theory and asymptotic analysis, to encounter expressions involving differences or ratios of harmonic numbers at scaled indices. The identity involving the limiting difference between harmonic numbers at scaled indices and its relationship to the logarithmic function provides an intriguing example of how discrete sequences can asymptotically relate to continuous functions. This identity is expressed as

in the limit) reflects how summation over increasing segments of the harmonic series exhibits integral properties-th harmonic number, defined as

The harmonic numbers are a fundamental sequence in number theory and analysis, known for their logarithmic growth. This result leverages the fact that the sum of the inverses of integers (i.e., harmonic numbers) can be closely approximated by the natural logarithm function, plus a constant, especially when extended over large intervals., precludes the possibility of the harmonic series approaching a finite limit, thus providing a clear mathematical articulation of its divergence. The technique of approximating sums by integrals (specifically using the integral test or by direct integral approximation) is fundamental in deriving such results. This specific identity can be a consequence of these approximations, considering:

Harmonic limit derivation

:

This can be estimated using the integral test for convergence, or more directly by comparing it to the integral:

implies this minimally discrete region.

Double series formula

The harmonic number difference is an extension of the classic, alternating identity:

which can be generalized as the double series over the residues:

is the principal ideal (i.e., balancing each term with the modulus) reduces the magnitude of each term's contribution, ensuring convergence increases. For example:

functions similarly to telescoping over a very large sequence. It helps to flatten out the discrepancies that might otherwise lead to divergent behavior in a straightforward harmonic series. Note that the structure of the summands of this formula matches those of the interpolated harmonic number when both the domain and range). However, the interpretation and roles of the variables differ.

Deveci's Proof

A fundamental feature of the proof is the accumulation of the subtrahends into a unit fraction, where the extrema otherwise being implicit in the latter case due to the structural requirements of the proof. Since the cardinality, not the cardinality (which is ambiguous is the limit of the sliding window (the scaled, truncated series):

Integrals of logarithmic functions

To remember higher integrals, it is convenient to define:

is the nth harmonic number:

Then:

Approximating large numbers

The identities of logarithms can be used to approximate large numbers. Note that , where a, b, and c are arbitrary constants. Suppose that one wants to approximate the 44th Mersenne prime, . To get the base-10 logarithm, we would multiply 32,582,657 by , getting . We can then get .

Similarly, factorials can be approximated by summing the logarithms of the terms.

Complex logarithm identities

The complex logarithm is the complex number analogue of the logarithm function. No single valued function on the complex plane can satisfy the normal rules for logarithms. However, a multivalued function can be defined which satisfies most of the identities. It is usual to consider this as a function defined on a Riemann surface. A single valued version, called the principal value of the logarithm, can be defined which is discontinuous on the negative x axis, and is equal to the multivalued version on a single branch cut.

Definitions

In what follows, a capital first letter is used for the principal value of functions, and the lower case version is used for the multivalued function. The single valued version of definitions and identities is always given first, followed by a separate section for the multiple valued versions.

• is the standard natural logarithm of the real number .

• is the principal value of the arg function; its value is restricted to . It can be computed using .

• is the principal value of the complex logarithm function and has imaginary part in the range .

The multiple valued version of is a set, but it is easier to write it without braces and using it in formulas follows obvious rules.

• is the set of complex numbers v which satisfy

• is the set of possible values of the arg function applied to z.

When k is any integer:

Constants

Principal value forms:

Multiple value forms, for any k an integer:

Summation

Principal value forms:

Multiple value forms:

Powers

A complex power of a complex number can have many possible values.

Principal value form:

Multiple value forms:

Where , are any integers:

Asymptotic identities

Pronic numbers

As a consequence of the harmonic number difference, the natural logarithm is asymptotically approximated by a finite series difference, representing a truncation of the integral:

is the th triangular number is the sum of the first n even integers. Since the th pronic number is asymptotically equivalent to the th perfect square, it follows that:

Prime number theorem

The prime number theorem provides the following asymptotic equivalence:

is the prime counting function. This relationship is equal to:

is the harmonic mean asymptotically approaches a small constant. This behavior can also be derived from the properties of logarithmsth of the series is expressed as follows:

which generalizes to:

and:

as a unit magnitude forms a divergent series.

Real Arguments

These approximations extend to the real-valued domain through the interpolated harmonic number:

Stirling numbers

The natural logarithm is asymptotically related to the harmonic numbers by the Stirling numbers and the Gregory coefficients. in terms of Stirling numbers of the first kind:

See also

References

External links

• Logarithm in Mathwords