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Logarithmically concave function. In convex analysis, a non-negative function f : Rn → R+ is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality. for all x,y ∈ dom f and 0 < θ < 1.
In mathematics, the logarithm to base b is the inverse function of exponentiation with base b. That means that the logarithm of a number x to the base b is the exponent to which b must be raised to produce x. For example, since 1000 = 103, the logarithm base of 1000 is 3, or log10 (1000) = 3.
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].
Logarithmic scale. A logarithmic scale (or log scale) is a method used to display numerical data that spans a broad range of values, especially when there are significant differences between the magnitudes of the numbers involved. Unlike a linear scale where each unit of distance corresponds to the same increment, on a logarithmic scale each ...
Stirling's approximation. 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 natural logarithm of x is generally written as ln x, loge x, or sometimes, if the base e is implicit, simply log x. [2][3] Parentheses are sometimes added for clarity, giving ln (x), loge(x), or log (x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.
The logarithm of 2 in other bases is obtained with the formula. The common logarithm in particular is (OEIS: A007524) log 10 2 ≈ 0.301 029 995 663 981 195. {\displaystyle \log _ {10}2\approx 0.301\,029\,995\,663\,981\,195.} The inverse of this number is the binary logarithm of 10: (OEIS: A020862). By the Lindemann–Weierstrass theorem ...
Binary logarithm. Graph of log2 x as a function of a positive real number x. In mathematics, the binary logarithm (log2 n) is the power to which the number 2 must be raised to obtain the value n. That is, for any real number x, For example, the binary logarithm of 1 is 0, the binary logarithm of 2 is 1, the binary logarithm of 4 is 2, and the ...