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The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718 281 828 459. [1] The natural logarithm of x is generally written as ln x, loge x, or sometimes, if the base e is implicit, simply log x.
This form immediately generalizes to more alternatives as the softmax function, which is a vector-valued function whose i-th coordinate is / =. More subtly, the symmetric form emphasizes interpreting the input x as x 1 − x 0 {\displaystyle x_{1}-x_{0}} and thus relative to some reference point, implicitly to x 0 = 0 {\displaystyle x_{0}=0} .
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, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula. where is the derivative of f. [1] Intuitively, this is the infinitesimal relative change in f; that is, the infinitesimal absolute change in f, namely scaled by the current value of f.
Comparison of Linear, Concave, and Convex Functions In original (left) and log10 (right) scales. In science and engineering, a log–log graph or log–log plot is a two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Power functions – relationships of the form – appear as straight ...
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The following is a list of integrals ( antiderivative functions) of logarithmic functions. For a complete list of integral functions, see list of integrals . Note: x > 0 is assumed throughout this article, and the constant of integration is omitted for simplicity.
Von Mangoldt function, Λ(n) = log p if n is a positive power of the prime p; Carmichael function; Antiderivatives of elementary functions. Logarithmic integral function: Integral of the reciprocal of the logarithm, important in the prime number theorem. Exponential integral; Trigonometric integral: Including Sine Integral and Cosine Integral