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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.
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 ...
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 ...
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.
A stronger, related result is the five exponentials theorem, [4] which is as follows. Let x 1, x 2 and y 1, y 2 be two pairs of complex numbers, with each pair being linearly independent over the rational numbers, and let γ be a non-zero algebraic number. Then at least one of the following five numbers is transcendental:
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.
Binary logarithms are also used in computer science, where the binary system is ubiquitous; in music theory, where a pitch ratio of two (the octave) is ubiquitous and the number of cents between any two pitches is a scaled version of the binary logarithm, or log 2 times 1200, of the pitch ratio (that is, 100 cents per semitone in conventional ...
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 ...
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