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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].
The area interpretation allows the easy derivation of some basic properties of the logarithmic mean. Since the exponential function is monotonic, the integral over an interval of length 1 is bounded by x and y. The homogeneity of the integral operator is transferred to the mean operator, that is (,) = (,).
As an integral, ln(t) equals the area between the x-axis and the graph of the function 1/x, ranging from x = 1 to x = t. This is a consequence of the fundamental theorem of calculus and the fact that the derivative of ln(x) is 1/x. Product and power logarithm formulas can be derived from this definition. [41]
For example, ln i = iπ / 2 or 5iπ / 2 or - 3iπ / 2 , etc.; and although i 4 = 1, 4 ln i can be defined as 2iπ, or 10iπ or −6iπ, and so on. Plots of the natural logarithm function on the complex plane ( principal branch )
A probability distribution is not uniquely determined by the moments E[X n] = e nμ + 1 / 2 n 2 σ 2 for n ≥ 1. That is, there exist other distributions with the same set of moments. [4] In fact, there is a whole family of distributions with the same moments as the log-normal distribution. [citation needed]
Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative. [ citation needed ] Logarithms can be used to remove exponents, convert products into sums, and convert division into subtraction—each of which may lead to a simplified ...
The formula was first discovered by Abraham de Moivre [2] in the form ! [] +. De Moivre gave an approximate rational-number expression for the natural logarithm of the constant. Stirling's contribution consisted of showing that the constant is precisely 2 π {\displaystyle {\sqrt {2\pi }}} .
The fractional part of the result is log 2 y and can be computed iteratively, using only elementary multiplication and division. [58] The algorithm for computing the fractional part can be described in pseudocode as follows: Start with a real number y in the half-open interval [1, 2). If y = 1, then the algorithm is done, and the fractional ...