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In calculus, logarithmic differentiation or differentiation by taking logarithms is a method used to differentiate functions by employing the logarithmic derivative of a function f, [1] () ′ = ′ ′ = () ′.
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 f ′ , {\displaystyle f',} scaled by the current ...
In algebraic geometry and the theory of complex manifolds, a logarithmic differential form is a differential form with poles of a certain kind. The concept was introduced by Pierre Deligne . [ 1 ] In short, logarithmic differentials have the mildest possible singularities needed in order to give information about an open submanifold (the ...
Logarithms can be used to make calculations easier. For example, two numbers can be multiplied just by using a logarithm table and adding. These are often known as logarithmic properties, which are documented in the table below. [2] The first three operations below assume that x = b c and/or y = b d, so that log b (x) = c and log b (y) = d.
The notation convention chosen here (with W 0 and W −1) follows the canonical reference on the Lambert W function by Corless, Gonnet, Hare, Jeffrey and Knuth. [3]The name "product logarithm" can be understood as follows: since the inverse function of f(w) = e w is termed the logarithm, it makes sense to call the inverse "function" of the product we w the "product logarithm".
The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule): () ′ = ′, wherever is positive. Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative.
In algebraic geometry, a log structure provides an abstract context to study semistable schemes, and in particular the notion of logarithmic differential form and the related Hodge-theoretic concepts. This idea has applications in the theory of moduli spaces, in deformation theory and Fontaine's p-adic Hodge theory, among others.
If we consider functions of the form () = where is any complex number and is a complex number in a slit complex plane that excludes the branch point of 0 and any branch cut connected to it, and we use the conventional multivalued definition := (), then it is straightforward to show that, on each branch of the complex logarithm, the same ...