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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.
Logarithms: the inverses of exponential functions; useful to solve equations involving exponentials. Natural logarithm; Common logarithm; Binary logarithm; Power functions: raise a variable number to a fixed power; also known as Allometric functions; note: if the power is a rational number it is not strictly a transcendental function. Periodic ...
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 ...
One case of non-homogeneous quadratic relations is covered by the still open three exponentials conjecture. [10] In its logarithmic form it is the following conjecture. Let λ 1, λ 2, and λ 3 be any three logarithms of algebraic numbers and γ be a non-zero algebraic number, and suppose that λ 1 λ 2 = γλ 3. Then λ 1 λ 2 = γλ 3 = 0.
Baker's Theorem — If , …, are linearly independent over the rational numbers, then for any algebraic numbers , …,, not all zero, we have | + + + | > where H is the maximum of the heights of and C is an effectively computable number depending on n, and the maximum d of the degrees of . (If β 0 is nonzero then the assumption that are linearly independent can be dropped.)
If F is a commutative n-dimensional formal group law over a commutative Q-algebra R, then it is strictly isomorphic to the additive formal group law. [4] In other words, there is a strict isomorphism f from the additive formal group to F, called the logarithm of F, so that f(F(x,y)) = f(x) + f(y). Examples: The logarithm of F(x,y) = x + y is f ...
Sinkhorn's theorem (matrix theory) Sion's minimax theorem (game theory) Sipser–Lautemann theorem (probabilistic complexity theory) (structural complexity theory) Siu's semicontinuity theorem (complex analysis) Six circles theorem ; Six exponentials theorem (transcendental number theory) Sklar's theorem
2.2 Exponential function. 2.3 Trigonometric, inverse trigonometric, ... 7.5 Exponential and logarithms. 8 See also. 9 Notes. 10 References. Toggle the table of contents.
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