List of logarithmic identities
- 1 Trivial identities
- 2 Canceling exponentials
- 3 Using simpler operations
- 4 Changing the base
- 5 Inequalities
- 6 Calculus identities
- 7 Approximating large numbers
- 8 Complex logarithm identities
- 9 See also
- 10 References
- 11 External links
|because||, given that b>0|
Note that logb(0) is undefined because there is no number x such that bx = 0. In fact, there is a vertical asymptote on the graph of logb(x) at x = 0.
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).
Both of the above are derived from the following two equations that define a logarithm:-
Substituting c in the left equation gives blogb(x) = x, and substituting x in the right gives logb(bc) = c. Finally, replace c by x.
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. The first three operations below assume x = bc, and/or y = bd so that logb(x) = c and logb(y) = d. Derivations also use the log definitions x = blogb(x) and x = logb(bx).
Where , , and are positive real numbers and . Both and are real numbers.
The laws result from canceling exponentials and 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:
Changing the base
This identity is useful to evaluate logarithms on calculators. For instance, most calculators have buttons for ln and for log10, but not for log2. To find log2(3), one could calculate log10(3) / log10(2) (or ln(3)/ln(2), which yields the same result.
- Let .
- Then .
- Take on both sides:
- Simplify and solve for :
- Since , then
This formula has several consequences:
where 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:
Note that in practice and have to be switched on the right hand side of the equations if . Also note that the subtraction identity is not defined if since the logarithm of zero is not defined. Many programming languages have a specific
log1p(x) function that calculates without underflow when is small.
where are sorted in descending order.
A useful identity involving exponents:
Both are pretty sharp around x=0, but not for large x.
The last limit is often summarized as "logarithms grow more slowly than any power or root of x".
Derivatives of logarithmic functions
Where , , and .
To remember higher integrals, it's convenient to define:
Where is the nth Harmonic number.
Approximating large numbers
The identities of logarithms can be used to approximate large numbers. Note that logb(a) + logb(c) = logb(ac), where a, b, and c are arbitrary constants. Suppose that one wants to approximate the 44th Mersenne prime, 232,582,657 − 1. To get the base-10 logarithm, we would multiply 32,582,657 by log10(2), getting 9,808,357.09543 = 9,808,357 + 0.09543. We can then get 109,808,357 × 100.09543 ≈ 1.25 × 109,808,357.
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 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.
- ln(r) is the standard natural logarithm of the real number r.
- Log(z) is the principal value of the complex logarithm function and has imaginary part in the range (-π, π].
- Arg(z) is the principal value of the arg function, its value is restricted to (-π, π]. It can be computed using Arg(x+iy)= atan2(y, x).
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.
- log(z) is the set of complex numbers v which satisfy ev = z
- arg(z) is the set of possible values of the arg function applied to z.
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 k1, k2 are any integers: