Definitions

1. The logarithm is the mathematical operation that is the inverse of exponentiation (raising a constant, the base, to a power).
2. The word "logarithm" is often used to refer to a logarithm function itself as well as to particular values of this function.
3. The logarithm is a mathematical function much the same as more familiar functions such as the square root, sin, or absolute value functions.
4. The logarithm is one of three closely related functions.
5. The logarithm is the mathematical operation that is the inverse of exponentiation, or raising a number (the base) to a power.

Common Logarithm

1. See common logarithm for details, including the use of characteristics and mantissas of common (i.e., base-10) logarithms.
2. To compute powers or roots of a number, the common logarithm of that number was looked up and multiplied or divided by the radix.
3. Alternatively below is a correct but inefficient algorithm that can calculate the common logarithm to an arbitrary number of decimal places.
4. The logarithm to base b = 10 is called common logarithm, while the base b = 2 gives rise to the binary logarithm.
5. There is a curious approximation to the common logarithm that can be made on such a calculator.

Inverse Function

1. Also note that is its own inverse function, so to yield the natural logarithm of a certain number n, simply put in for x.
2. See logarithmic identities for several rules governing the logarithm functions.
3. For example, the logarithm of a matrix is the (multi-valued) inverse function of the matrix exponential.
4. The logarithm for a base and a number is defined to be the inverse function of taking to the power, i.e.,.

Logarithm Function

1. The logarithm function can be extended to a Haar measure in the topological group of positive real numbers under multiplication.
2. The notation Log( x) is also used by mathematicians to denote the principal branch of the (natural) logarithm function.
3. Similarly, the logarithm function can be defined for any positive real number.
4. The notation Log( x) is also used by mathematicians to denote the principal branch of the (natural) logarithm function.
5. The logarithm function is simply another rule or method used to match one real number with another.
6. Here, as we did above for real exponents, we used the natural logarithm function ln to write .

Logarithms

1. Arithmetical complement of a logarithm, the difference between a logarithm and the number ten.
2. The derivative of the logarithm of the Gamma function is called the digamma function; higher derivatives are the polygamma functions.
3. IV., article "Prony.") Cubic interpolation could be used to find the logarithm of any number to a similar accuracy.
4. It is possible to take the logarithm of a quaternions and octonions.
5. For more information on logarithm, visit Britannica.com.

Discrete Logarithm

1. For some finite groups, it is believed that the discrete logarithm is very hard to calculate, whereas discrete exponentials are quite easy.
2. The discrete logarithm is a related notion in the theory of finite groups.

Natural Logarithm

1. Extending the natural logarithm to complex arguments yields a multi-valued function, ln( z).
2. The neper is a similar unit which uses the natural logarithm of a ratio.
3. Mathematicians, on the other hand, wrote "log( x)" when they mean log e( x) for the natural logarithm.
4. This is explained on the natural logarithm page.
5. Neither the natural logarithm nor the square root function is entire.

Common

1. Computers Most computer languages use log( x) for the natural logarithm, while the common log is typically denoted log10(x).
2. Alternatively below is a correct but inefficient algorithm that can calculate the common logarithm to an arbitrary number of decimal places.
3. On calculators it is usually "log", but mathematicians usually mean natural logarithm rather than common logarithm when they write "log".
4. There is a curious approximation to the common logarithm that can be made on such a calculator.
5. Positional notation contains a simple logarithmic function because the position is the common logarithm of the corresponding weight.

Number

1. IV., article "Prony.") Cubic interpolation could be used to find the logarithm of any number to a similar accuracy.
2. As an example, there are a number of simple series involving the natural logarithm.
3. In words, to raise a number to a power p, find the logarithm of the number and multiply it by p.
4. Negative logarithm values were rarely converted to a normal negative number (−0.920819 in the example).
5. For the logarithm to be defined, the base b must be a positive real number not equal to 1 and y must be a positive number.

Super-Logarithm

1. The super-logarithm of x grows even more slowly than the double logarithm for large x.
2. A super-logarithm or hyper -4-logarithm is the inverse function of tetration.

Finite Groups

1. For some finite groups, it is believed that the discrete logarithm is very hard to calculate, whereas discrete exponentials are quite easy.
2. The discrete logarithm is a related notion in the theory of finite groups.

Inverse

1. A double logarithm is the inverse function of the double-exponential function.
2. A super-logarithm or hyper-logarithm is the inverse function of the super-exponential function.
3. An antilogarithm is used to show the inverse of the logarithm.
4. This is called finding the antilogarithm or inverse logarithm of the number.
5. The exponential function, exp x, may be defined as the inverse of the logarithm: thus x =exp y if y= log x.

Base

1. It is also known as the decadic logarithm, named after its base.
2. To compute a base 2 logarithm on a number between 1 and 2 in an alternate way, square it repeatedly.
3. In German, lg( x) also denotes the base 10 logarithm, while sometimes ld( x) or lb( x) is used for the base 2 logarithm.
4. However, in Russian literature, the notation lg( x) is generally used for the base 10 logarithm, so even this usage is not without its perils.
5. The logarithm base 10 of a number.

Exponential

1. Since log is the logarithm base 10, we apply the exponential function base 10 to both sides of the equation.
2. In fact, if this method is used, Newton inversion of the natural logarithm may conversely be used to calculate the exponential function efficiently.
3. For a given, constant base such as e or 10, the exponential function "undoes" the logarithm function, and the logarithm undoes the exponential.
4. Viewed in this way, the base-b logarithm function is the inverse function of the exponential function b x.
5. This exponential function can be inverted to form a complex logarithm that exhibits most of the properties of the ordinary logarithm.

Double Logarithm

1. The super-logarithm of x grows even more slowly than the double logarithm for large x.
2. The super-logarithm of x grows even slower than the double logarithm for large x.

Means

1. Mathematicians generally understand both "ln( x)" and "log( x)" to mean log e( x) and write "log 10( x)" when the base-10 logarithm of x is intended.
2. The computational complexity of computing the natural logarithm (using the arithmetic-geometric mean) is O( M( n) ln n).

Base Logarithm

1. However, in Russian literature, the notation lg( x) is generally used for the base 10 logarithm, so even this usage is not without its perils.[1].
2. To compute a base 2 logarithm on a number between 1 and 2 in an alternate way, square it repeatedly.
3. In computer science, the base 2 logarithm is sometimes written as lg( x) to avoid confusion.
4. For 10 as the base of the logarithm we've got the common logarithm log(x), also called Briggs' logarithm or Briggian logarithm.
5. One's choice of base with logarithms is not crucial, because a logarithm can be converted from one base to another quite easily.

Positive Real Numbers

1. The logarithm function can be extended to a Haar measure in the topological group of positive real numbers under multiplication.
2. Similarly, the logarithm function can be defined for any positive real number.
3. In other words, the logarithm function is a bijection from the set of positive real numbers to the set of all real numbers.

Complex

1. The complex logarithm can only be single-valued on the cut plane.
2. This exponential function can be inverted to form a complex logarithm that exhibits most of the properties of the ordinary logarithm.

Rule

1. See logarithmic identities for several rules governing the logarithm functions.
2. The same identity is exploited when computing square roots with logarithm tables or slide rules.

Used

1. In geometry the logarithm is used to form the metric for the half-plane model of hyperbolic geometry.
2. In most commonly used programming languages, including C, C++, Fortran, and BASIC, "log" or "LOG" means natural logarithm.
3. In most commonly used computer programming languages, including C, C++, Java, Fortran, and BASIC, the "log" or "LOG" function returns the natural logarithm.

Power

1. The bel (symbol B) is a unit of measure which is the base-10 logarithm of ratios, such as power levels and voltage levels.
2. To compute powers or roots of a number, the common logarithm of that number was looked up and multiplied or divided by the radix.
3. In mathematics, a logarithm of a given number to a given base is the power to which you need to raise the base in order to get the number.
4. In most high school and lower division college courses, log x is the common logarithm: the power to which ten must be raised to get x.
5. The three parts of a logarithm are a base, an argument (also called power) and an answer.

Logarithmic

1. In mathematics, the logarithmic integral function or integral logarithm li( x) is a special function.
2. The logarithm functions are the inverses of the exponential functions.
3. In mathematics, a logarithm of x with base b may be defined as the following: for the equation b n = x, the logarithm is a function which gives n.
4. Graph of the natural logarithm function.
5. The natural logarithm is a logarithmic function that takes the exponential, e, as its base.

Written

1. It is written as exp( x) or e x, where e is the base of the natural logarithm.
2. In computer science, the base 2 logarithm is sometimes written as lg( x) to avoid confusion.
3. In computer science, the base 2 logarithm is sometimes written as lg( x), as suggested by Edward Reingold and popularized by Donald Knuth.
4. The logarithm of x to the base b is written log b( x) or, if the base is implicit, as log( x).
5. In decadic (common), or Briggsian, logarithms the base is 10; the common logarithm of a is written as log 10 a, or lg a.

Including

1. In most commonly used computer programming languages, including C, C++, Java, Fortran, Ruby, and BASIC, the "log" function returns the natural logarithm.
2. See common logarithm for details, including the use of characteristics and mantissa s of common (i.e., base-10) logarithms.

Operations

1. In addition to being an isomorphism of sets, the logarithm function also preserves certain operations.
2. Image:Logarithms.png The logarithm is the mathematical operation that is the inverse of exponentiation, or raising a number (the base) to a power.

Indefinite Logarithm

1. In the case of the quantities that result from the indefinite logarithm function, their associated units are called logarithmic units.
2. One satisfactory mathematical definition of the indefinite logarithm operator Log, using lambda calculus notation, is Log = -- x.-- b.log b x.

Logarithm Base

1. This function is a group isomorphism, called the discrete logarithm to base b.
2. It is also known as the decadic logarithm, named after its base.
3. Moreover, this result implies that all logarithm functions (whatever the base) are similar to each other.
4. The numerical value for logarithm to the base 10 can be calculated with the following identity.
5. If n x = a, the logarithm of a, with n as the base, is x; symbolically, log n a = x.

Similar

1. Moreover, this result implies that all logarithm functions (whatever the base) are similar to each other.
2. The neper is a similar unit which uses the natural logarithm of a ratio.

Calculated

1. The numerical value for logarithm to the base 10 can be calculated with the following identity.
2. Thus if is a number and is its logarithm as calculated by Napier, .

Inverse Logarithm

1. The logarithm of a matrix is the inverse of the matrix exponential.
2. An antilogarithm is used to show the inverse of the logarithm.
3. Complex Variables The complex logarithm is the inverse of the complex exponential function.
4. The antilogarithm is the logarithmic inverse of the logarithm, meaning that the antilog b(log b(x))=x.
5. The exponentiated value is then the inverse logarithm of this product; that is, number to power = b product.

Logarithm

1. Mathematicians generally understand both "ln( x)" and "log( x)" to mean log e( x) and write "log 10( x)" when the base-10 logarithm of x is intended.
2. Viewed on a graph, the Digamma Function resembles the tangent function for z < 0 and a less steep logarithm ic function for z 0.
3. The bel (symbol B) is a unit of measure which is the base-10 logarithm of ratios, such as power levels and voltage levels.
4. The function e x so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e.
5. The complex logarithm is the inverse of the exponential function applied to complex numbers and generalizes the logarithm to complex numbers.

Multiplication

1. Also, the multiplication factor used to convert a logarithm of one base to a logarithm of another base.

Solution

1. In particular, an ordinary logarithm log a(b) is a solution of the equation a x  =  b over the real or complex numbers.
2. The solution also applies for where the logarithm is not defined.

Sum

1. Returns the natural logarithm of the sum of the argument and 1.

Equal

1. Note that in particular for f 2 if we set c equal to e, the base of the natural logarithm, then we get that e x is a transcendental function.

Times

1. It is based on the common logarithm of ratios —10 times the common logarithm of a power ratio or 20 times the common logarithm of a voltage ratio.
2. Usually, it is ten times the base-10 logarithm of the ratio.
3. So the logarithm function is in fact a group isomorphism from the group (mathbb{R}^+,times) to the group (mathbb{R},+).

Principal Value

1. However, the above formulas for logarithms of products and powers do not generalize to the principal value of the complex logarithm.

Argument

1. The `log10(x)` function returns the logarithm (base 10) of its argument.
2. The `log(x)` function returns the natural logarithm (base `e`) of its argument.

Solving

1. In the present invention, the encryption of data is performed relying on the difficulty in solving the discrete logarithm problem.

Notation

1. So a logarithm of z is a complex number w such that e w = z.[ 1] The notation for such a w is log z.
2. The notation "ln(x)" invariably means log e(x), i.e., the natural logarithm of x.
3. The notation is used by physicists, engineers, and calculator keypads to denote the common logarithm.

Quantity

1. The factor of 20 comes from the fact that the logarithm of the square of a quantity is equal to 2 x the logarithm of the quantity.

Calculators

1. Some calculators might only have base 10 logarithms and base e logarithms (natural logarithm), so to evaluate a base 5 log you will need to change the base.
2. Before calculators became available, via logarithm tables, logarithms were crucial to simplifying scientific calculations.
3. Notes: Some calculators, of course, allow you to extract the logarithm of any number to any base.

Calculator

1. This is yet another reason why (from the 1600s to the era of the calculator) mathematicians and scientists usually use the natural log (e-based) logarithm.

Powers

1. The first is about the logarithm of a product, the second about logarithms of powers and the third involves logarithms with respect to different bases.

Variable

1. Therefore the logarithm only depends on the variable x, a positive real number.

Complex Plane

1. In case of α i a complex number, one has to allow log to denote some definite branch of the logarithm function in the complex plane.

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