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ln (r) is the standard natural logarithm of the real number r. 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). Log (z) is the principal value of the complex logarithm function and has imaginary part in the range (−π, π].
In mathematics, the logarithmto basebis the inverse function of exponentiationwith base b. That means that the logarithm of a number xto the base bis the exponentto which bmust be raised to produce x. For example, since 1000 = 103, the logarithm base 10{\displaystyle 10}of 1000is 3, or log10 (1000) = 3.
An important property of base-10 logarithms, which makes them so useful in calculations, is that the logarithm of numbers greater than 1 that differ by a factor of a power of 10 all have the same fractional part. The fractional part is known as the mantissa. [note 2] Thus, log tables need only show the fractional part. Tables of common ...
The natural logarithm function, if considered as a real-valued function of a positive real variable, is the inverse function of the exponential function, leading to the identities: = + = Like all logarithms, the natural logarithm maps multiplication of positive numbers into addition: [ 5 ] ln ( x ⋅ y ) = ln x + ln y ...
Stirling's approximation. Appearance. hide. Comparison of Stirling's approximation with the factorial. In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of .
The term Napierian logarithm or Naperian logarithm, named after John Napier, is often used to mean the natural logarithm. Napier did not introduce this natural logarithmic function, although it is named after him. [1][2] However, if it is taken to mean the "logarithms" as originally produced by Napier, it is a function given by (in terms of the ...
Discrete logarithm. In mathematics, for given real numbers a and b, the logarithm log b a is a number x such that bx = a. Analogously, in any group G, powers bk can be defined for all integers k, and the discrete logarithm log b a is an integer k such that bk = a. In number theory, the more commonly used term is index: we can write x = ind r a ...
The function , for x < 107. In mathematics, the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev functionϑ(x) or θ(x) is given by. where denotes the natural logarithm, with the sum extending over all prime numbers p that are less than or equal to x.