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Logarithms can be used to make calculations easier. For example, two numbers can be multiplied just by using a logarithm table and adding. Where formula_1, formula_2, and formula_3 are positive real numbers and formula_4. Both formula_5 and formula_6 are real numbers.
Note that formula_7 is undefined because there is no number formula_8 such that formula_9. In fact, there is a vertical asymptote on the graph of formula_10 at formula_11.
Logarithms and exponentials (antilogarithms) with the same base cancel each other. This is true because logarithms and exponentials are inverse operations (just like multiplication and division or addition and subtraction).
This identity is needed to evaluate logarithms on calculators. For instance, most calculators have buttons for ln and for log, but not for log. To find log(3), one must calculate log(3) / log(2) (or ln(3)/ln(2), which yields the same result).
This formula has several consequences:
where formula_26 is any permutation of the subscripts 1, ..., "n". For example
The following summation/subtraction rule is especially useful in probability theory when one is dealing with a sum of log-probabilities:
which gives the special cases:
Note that in practice formula_32 and formula_5 have to be switched on the right hand side of the equations if formula_34. Also note that the subtraction identity is not defined if formula_35 since the logarithm of zero is not defined.
More generally:
where formula_37.
A useful identity involving exponents:
The last limit is often summarized as "logarithms grow more slowly than any power or root of "x"".
Where formula_47, formula_48, and formula_4.
To remember higher integrals, it's convenient to define:
Then,
The identities of logarithms can be used to approximate large numbers. Note that log("a") + log("c") = log("ac"), where "a", "b", and "c" are arbitrary constants. Suppose that one wants to approximate the 44th Mersenne prime, 2 − 1. To get the base-10 logarithm, we would multiply 32,582,657 by log(2), getting 9,808,357.09543 = 9,808,357 + 0.09543. We can then get 10 × 10 ≈ 1.25 × 10.
Similarly, factorials can be approximated by summing the logarithms of the terms.
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 equals the multivalued version on a single branch cut.
The convention will be used here that a capital first letter is used for the principal value of functions and the lower case version refers to 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.
The multiple valued version of log("z") is a set but it is easier to write it without braces and using it in formulas follows obvious rules.
When "k" is any integer:
Principal value forms:
Multiple value forms, for any "k" an integer:
Principal value forms:
Multiple value forms:
A complex power of a complex number can have many possible values.
Principal value form:
Multiple value forms:
Where "k", "k" are any integers:
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