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The graph of the logarithm base 2 crosses the x-axis at x = 1 and passes through the points (2, 1), (4, 2), and (8, 3), depicting, e.g., log 2 (8) = 3 and 2 3 = 8. The graph gets arbitrarily close to the y-axis, but does not meet it. Addition, multiplication, and exponentiation are three of the most fundamental arithmetic operations.
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. The exponential form of this conjecture is the following. Let x 1, x 2, and y be non-zero complex numbers and let γ be a non-zero algebraic number. Then at least one of ...
The identities of logarithms can be used to approximate large numbers. Note that log b (a) + log b (c) = log b (ac), where a, b, and c are arbitrary constants. Suppose that one wants to approximate the 44th Mersenne prime, 2 32,582,657 −1. To get the base-10 logarithm, we would multiply 32,582,657 by log 10 (2), getting 9,808,357.09543 ...
In 2017, it was proven [14] that there exists a unique function F which is a solution of the equation F(z + 1) = exp(F(z)) and satisfies the additional conditions that F(0) = 1 and F(z) approaches the fixed points of the logarithm (roughly 0.318 ± 1.337i) as z approaches ±i∞ and that F is holomorphic in the whole complex z-plane, except the ...
3) Systems of Equations 3.1 Systems of Linear Equations in Three Variables; 3.2 Simultaneous Equations involving One Linear Equation and One Non-Linear Equations; 4) Indices, Surds and Logarithms 4.1 Law of Indices; 4.2 Laws of Surds; 4.3 Laws of Logarithms; 4.4 Applications of Indices, Surds and Logarithms; 5) Progressions 5.1 Arithmetic ...
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 ...
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