The logarithmic derivative of a function is defined as the derivative of the logarithm of a function. For example, the digamma function is defined as the logarithmic derivative of the gamma function,
For all integers and nonnegative integers , the harmonic logarithms of order and degree are defined as the unique functions satisfying 1. , 2. has no constant term except , 3. , where the "Roman symbol" is defined by(1)(Roman 1992). This gives the special cases(2)(3)where is a harmonic number. The harmonic logarithm has the integral(4)The harmonic logarithm can be written(5)where is the differential operator, (so is the th integral). Rearranging gives(6)This formulation gives an analog of the binomial theorem called the logarithmic binomial theorem. Another expression for the harmonic logarithm is(7)where is a Pochhammer symbol and is a two-index harmonic number (Roman 1992).
Min Max The common logarithm is the logarithm to base 10. The notation is used by physicists, engineers, and calculator keypads to denote the common logarithm. However, mathematicians generally use the same symbol to mean the natural logarithm ln, . Worse still, in Russian literature the notation is used to denote a base-10 logarithm, which conflicts with the use of the symbol lg to indicate the logarithm to base 2. To avoid all ambiguity, it is best to explicitly specify when the logarithm to base 10 is intended. In this work, , is used for the natural logarithm, and is used for the logarithm to the base 2.The situation is complicated even more by the fact that number theorists (e.g., Ivić 2003) commonly use the notation to denote the nested natural logarithm .The common logarithm is implemented in the Wolfram Language as Log[10, x] and Log10[x].Hardy and Wright (1979, p. 8) assert that the common logarithm has "no mathematical interest."Common..
The decimal expansion of the natural logarithm of 10 is given by(1)(OEIS A002392).It is also given by the BBP-type formulas(2)(3)(4)(5)(6)(7)(E. W. Weisstein, Aug. 28, 2008).
The logarithm for a base and a number is defined to be the inverse function of taking to the power , i.e., . Therefore, for any and ,(1)or equivalently,(2)For any base, the logarithm function has a singularity at . In the above plot, the blue curve is the logarithm to base 2 (), the black curve is the logarithm to base (the natural logarithm ), and the red curve is the logarithm to base 10 (the common logarithm, i.e., ).Note that while logarithm base 10 is denoted in this work, on calculators, and in elementary algebra and calculus textbooks, mathematicians and advanced mathematics texts uniformly use the notation to mean , and therefore use to mean the common logarithm. Extreme care is therefore needed when consulting the literature.The situation is complicated even more by the fact that number theorists (e.g., Ivić 2003) commonly use the notation to denote the nested natural logarithm .In the Wolfram Language, the logarithm to the base is implemented..
The log sine function, also called the logsine function, is defined by(1)The first few cases are given by(2)(3)(4)where is the Riemann zeta function.The log sine function is related to the log cosinefunction by(5)and the two are equal if the range of integration for is restricted from 0 to to 0 to .
The catacaustic of the natural logarithm specified parametrically as(1)(2)is a complicated expression for an arbitrary radiantpoint.However, for a point , the catacaustic becomes(3)(4)Making the substitution then gives the equivalent parametrization(5)(6)which is the equation of a catenary.
By analogy with the log sine function, definethe log cosine function by(1)The first few cases are given by(2)(3)(4)where is the Riemann zeta function.The log cosine function is related to the log sinefunction by(5)and the two are equal if the range of integration for is restricted from 0 to to 0 to .Oloa (2011) computed an exact value of the log cosine integral(6)where is the Riemann zeta function, is the Euler-Mascheroni constant, is a multivariate zeta function, and denotes . A closed form for in terms of more elementary functions is not known as of Apr. 2011, but it is numerically given by(7)(Oloa 2011; OEIS A189272).
The binary logarithm is the logarithm to base 2.The notation is sometimes used to denote this function in number theoretic literature. However, because Russian and German literature use the symbol to denote the base-10 logarithm and since this is the use recommended by the United States Department of Commerce (Taylor 1995, p. 33), this practice is discouraged. (To confuse matters even more, some German literature uses the notation to mean the binary logarithm.)The binary logarithm is implemented in the Wolfram Language as Log[2, z] and Log2[z].When information theoretic functions (like entropy) are computed using , the units of information are obtained in bits. When is used instead, the units of information are known as "nats."
The natural logarithm is the logarithm having base e, where(1)This function can be defined(2)for .This definition means that e is the unique number with the property that the area of the region bounded by the hyperbola , the x-axis, and the vertical lines and is 1. In other words,(3)The notation is used in physics and engineering to denote the natural logarithm, while mathematicians commonly use the notation . In this work, denotes a natural logarithm, whereas denotes the common logarithm.There are a number of notational conventions in common use for indication of a power of a natural logarithm. While some authors use (i.e., using a trigonometric function-like convention), it is also common to write .Common and natural logarithms can be expressed in terms of each other as(4)(5)The natural logarithm is especially useful in calculusbecause its derivative is given by the simple equation(6)whereas logarithms in other bases have the more complicated..
is the notation used in physics and engineering to denote the logarithm to base e, also called the natural logarithm, i.e.,The United States Department of Commerce recommends that the notation be used in this way to refer to the natural logarithm (Taylor 1995, p. 33).Unfortunately, mathematicians in the United States commonly use the symbol to refer to the natural logarithm, as does TraditionalForm typesetting in the Wolfram Language. The use of for different purposes by different mathematical communities causes considerable confusion, so extreme care is needed in determining if the symbol found in the wild refers to or .The natural logarithm is implemented in the Wolfram Language as Log[x], which is equivalent to Log[E, x].
The first definition of the logarithm was constructed by Napier and popularized through his posthumous pamphlet (Napier 1619). It this pamphlet, Napier sought to reduce the operations of multiplication, division, and root extraction to addition and subtraction. To this end, he defined the "logarithm" of a number by(1)written .This definition leads to the remarkable relations(2)(3)(4)which give the identities(5)(6)(7)(Havil 2003, pp. 8-9). While Napier's definition is different from the modern one (in particular, it decreases with increasing , but also fails to satisfy a number of properties of the modern logarithm), it provides the desired property of transforming multiplication into addition.The Napierian logarithm can be given in terms of the modern logarithm by solving equation (1) for , giving(8)Because a ratio of logarithms appears in this expression, any logarithm base can be used as long as the same value..
There are several conflicting meanings associated with the notation . In German and Russian literature, the notation is used to mean the common logarithm . This is also the usage recommended by the United States Department of Commerce (Taylor 1995, p. 33).However, is sometimes identified with the binary logarithm in some number theoretic literature (and here, mean the base-2 logarithm, not the nested natural logarithm as defined by Ivić 2003).Great care is therefore needed to determine the intended definition for when it is encountered in the wild.
The inverse function of the logarithm,defined such thatThe antilogarithm in base of is therefore .
The Mercator series, also called the Newton-Mercator series (Havil 2003, p. 33), is the Taylor series for the natural logarithm(1)(2)for , which was found by Newton, but independently discovered and first published by Mercator in 1668.Plugging in gives a beautiful series for the natural logarithm of 2,(3)also known as the alternating harmonic series and equal to , where is the Dirichlet eta function.
In number theory (e.g., Ivić 2003), the symbol is commonly used to mean the nested logarithm (also called the repeated logarithm or iterated logarithm) , where is the natural logarithm.Care must be taken based on context as to when the notation denotes the logarithm to base and when it means the -nested natural logarithm.The plots above show , , and in the complex plane.The penchant for formulas and bounds containing a profusion of nested logarithms has led to the following joke. What sound does a drowning analytic number theorist make? A: log log log log... (Havil 2003, p. 115).
The word "base" in mathematics is used to refer to a particular mathematical object that is used as a building block. The most common uses are the related concepts of the number system whose digits are used to represent numbers and the number system in which logarithms are defined. It can also be used to refer to the bottom edge or surface of a geometric figure.A real number can be represented using any integer number as a base (sometimes also called a radix or scale). The choice of a base yields to a representation of numbers known as a number system. In base , the digits 0, 1, ..., are used (where, by convention, for bases larger than 10, the symbols A, B, C, ... are generally used as symbols representing the decimal numbers 10, 11, 12, ...).The digits of a number in base (for integer ) can be obtained in the Wolfram Language using IntegerDigits[x, b].Let the base representation of a number be written(1)(e.g., ). Then, for example, the number 10 is..
The natural logarithm of 2 is a transcendental quantity that arises often in decay problems, especially when half-lives are being converted to decay constants. has numerical value(1)(OEIS A002162).The irrationality measure of is known to be less than 3.8913998 (Rukhadze 1987, Hata 1990).It is not known if is normal (Bailey and Crandall 2002).The alternating series and BBP-typeformula(2)converges to the natural logarithm of 2, where is the Dirichlet eta function. This identity follows immediately from setting in the Mercator series, yielding(3)It is also a special case of the identity(4)where is the Lerch transcendent.This is the simplest in an infinite class of such identities, the first few of which are(5)(6)(E. W. Weisstein, Oct. 7, 2007).There are many other classes of BBP-type formulas for , including(7)(8)(9)(10)(11)Plouffe (2006) found the beautiful sum(12)A rapidly converging Zeilberger-type sum..