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: , since , since
Explanations By definition, we know that:
where and .
Setting , we can see that:
So, substituting these values into the formula, we see that:
which gets us the first property.
Setting , 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:
Both of the above are derived from the following two equations that define a logarithm: (note that in this explanation, the variables of and may not be referring to the same number)
Looking at the equation , and substituting the value for of , we get the following equation:
which gets us the first equation.
Another more rough way to think about it is that , and that that "" is .
Looking at the equation , and substituting the value for of , we get the following equation:
which gets us the second equation.
Another more rough way to think about it is that , and that that "" is .
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 .
because because because because because because
Where , , and are positive real numbers and , and and 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:
Let , where , and let . We want to relate the expressions and . This can be done more easily by rewriting in terms of exponentials, whose properties we already know. Additionally, since we are going to refer to and quite often, we will give them some variable names to make working with them easier: Let , and let .
Rewriting these as exponentials, we see that
From here, we can relate (i.e. ) and (i.e. ) using exponent laws as
To recover the logarithms, we apply to both sides of the equality.
The right side may be simplified using one of the logarithm properties from before: we know that , giving
We now resubstitute the values for and into our equation, so our final expression is only in terms of , , and .
This completes the derivation.
Logarithm of a quotient
To state the logarithm of a quotient law formally:
Derivation:
Let , where , and let .
We want to relate the expressions and . This can be done more easily by rewriting in terms of exponentials, whose properties we already know. Additionally, since we are going to refer to and quite often, we will give them some variable names to make working with them easier: Let , and let .
Rewriting these as exponentials, we see that:
From here, we can relate (i.e. ) and (i.e. ) using exponent laws as
To recover the logarithms, we apply to both sides of the equality.
The right side may be simplified using one of the logarithm properties from before: we know that , giving
We now resubstitute the values for and into our equation, so our final expression is only in terms of , , and .
This completes the derivation.
Logarithm of a power
To state the logarithm of a power law formally:
Derivation:
Let , where , let , and let . For this derivation, we want to simplify the expression . To do this, we begin with the simpler expression . Since we will be using often, we will define it as a new variable: Let .
To more easily manipulate the expression, we rewrite it as an exponential. By definition, , so we have
Similar to the derivations above, we take advantage of another exponent law. In order to have in our final expression, we raise both sides of the equality to the power of :
where we used the exponent law .
To recover the logarithms, we apply to both sides of the equality.
The left side of the equality can be simplified using a logarithm law, which states that .
Substituting in the original value for , 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 section, the definition of a logarithm is: Solving for in the exponential equation could be done on most calculators by using common or natural logarithms: For example, the binary logarithm (), which is widely used in , could be calculated on most calculators as: More generally, the change of base formula can be formally defined as:
Proof and derivation
Let , where Let . Here, and 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. The number will be what the logarithm is evaluating, so it must be a positive number. Since we will be dealing with the term quite frequently, we define it as a new variable: Let .
To more easily manipulate the expression, it can be rewritten as an exponential.
Applying to both sides of the equality,
Now, using the logarithm of a power property, which states that ,
Isolating , we get the following:
Resubstituting back into the equation,
This completes the proof that .
This formula has several consequences:
where 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:
becausebecause
Note that the subtraction identity is not defined if , since the logarithm of zero is not defined. Also note that, when programming, and may have to be switched on the right hand side of the equations if to avoid losing the "1 +" due to rounding errors. Many programming languages have a specific log1p(x) function that calculates without underflow (when is small).
More generally:
Exponents A useful identity involving exponents:
or more universally:
Other or resulting identities
Inequalities Based on,
All are accurate around , 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
To modify the limits of integration to run from to , we change the order of integration, which changes the sign of the integral:
Therefore:
Riemann Sum
for and is a sample point in each interval.
Series representation
The natural logarithm has a well-known Taylor series expansion that converges for in the open-closed interval :
Within this interval, for , the series is conditionally convergent, and for all other values, it is absolutely convergent. For or , the series does not converge to . In these cases, different representationsTo extend the utility of the Mercator series beyond its conventional bounds one can calculate for and and then negate the result, , to derive . For example, setting yields . 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. See page 117, and VI.8 definition of shifted harmonic numbers on page 389 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 See Theorem 5.2. on pages 22 - 23
which characterizes the behavior of harmonic numbers as they grow large. This approximation (which precisely equals in the limit) reflects how summation over increasing segments of the harmonic series exhibits integral properties, giving insight into the interplay between discrete and continuous analysis. It also illustrates how understanding the behavior of sums and series at large scales can lead to insightful conclusions about their properties. Here denotes the -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. See formula 13. As tends towards infinity, the difference between the harmonic numbers and converges to a non-zero value. This persistent non-zero difference, , precludes the possibility of the harmonic series approaching a finite limit, thus providing a clear mathematical articulation of its divergence. See Proofs 23 and 24 for details on the relationship between harmonic numbers and logarithmic functions. 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
The limit explores the growth of the harmonic numbers when indices are multiplied by a scaling factor and then differenced. It specifically captures the sum from to :
This can be estimated using the integral test for convergence, or more directly by comparing it to the integral of from to :
As the window's lower bound begins at and the upper bound extends to , both of which tend toward infinity as , the summation window encompasses an increasingly vast portion of the smallest possible terms of the harmonic series (those with astronomically large denominators), creating a discrete sum that stretches towards infinity, which mirrors how continuous integrals accumulate value across an infinitesimally fine partitioning of the domain. In the limit, the interval is effectively from to where the onset implies this minimally discrete region.
Double series formula The harmonic number difference formula for is an extension of the classic, alternating identity of :
which can be generalized as the double series over the residues of :