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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 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.
[b] [6] [7] [8] Sometimes, it is called first-order propositional logic [9] to contrast it with System F, but it should not be confused with first-order logic. It deals with propositions [ 1 ] (which can be true or false ) [ 10 ] and relations between propositions, [ 11 ] including the construction of arguments based on them. [ 12 ]
In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable that can be written using only the basic operations of addition, subtraction, multiplication, and division (without the need of taking limits).
Similarly, f is strictly logarithmically convex if and only if, in the above two expressions, strict inequality holds for all t ∈ (0, 1). The above definition permits f to be zero, but if f is logarithmically convex and vanishes anywhere in X, then it vanishes everywhere in the interior of X.
If and are sufficiently small matrices, then can be computed as the logarithm of , where the exponentials and the logarithm can be computed as power series. The point of the Baker–Campbell–Hausdorff formula is then the highly nonobvious claim that Z := log ( e X e Y ) {\displaystyle Z:=\log \left(e^{X}e^{Y}\right)} can be expressed as a ...
In convex analysis, a non-negative function f : R n → R + is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality f ( θ x + ( 1 − θ ) y ) ≥ f ( x ) θ f ( y ) 1 − θ {\displaystyle f(\theta x+(1-\theta )y)\geq f(x)^{\theta }f(y)^{1-\theta }}
Moreover, as the derivative of f(x) evaluates to ln(b) b x by the properties of the exponential function, the chain rule implies that the derivative of log b x is given by [35] [37] = . That is, the slope of the tangent touching the graph of the base- b logarithm at the point ( x , log b ( x )) equals 1/( x ln( b )) .