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The logarithm of an infinite cardinal number κ is defined as the least cardinal number μ such that κ ≤ 2 μ. Logarithms of infinite cardinals are useful in some fields of mathematics, for example in the study of cardinal invariants of topological spaces, though they lack some of the properties that logarithms of positive real numbers possess.
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.
A list of articles about numbers (not about numerals). Topics include powers of ten, notable integers, prime and cardinal numbers, and the myriad system.
In mathematics, the logarithm to base b is the inverse function of exponentiation with base b. That means that the logarithm of a number x to the base b is the exponent to which b must be raised to produce x. For example, since 1000 = 10 3, the logarithm base of 1000 is 3, or log 10 (1000) = 3.
Notably, ℵ ω is the first uncountable cardinal number that can be demonstrated within Zermelo–Fraenkel set theory not to be equal to the cardinality of the set of all real numbers 2 ℵ 0: For any natural number n ≥ 1, we can consistently assume that 2 ℵ 0 = ℵ n, and moreover it is possible to assume that 2 ℵ 0 is as least as large ...
Existence of a cardinal number κ of a given type implies the existence of cardinals of most of the types listed above that type, and for most listed cardinal descriptions φ of lesser consistency strength, V κ satisfies "there is an unbounded class of cardinals satisfying φ".
The logarithm in the table, however, is of that sine value divided by 10,000,000. [1]: p. 19 The logarithm is again presented as an integer with an implied denominator of 10,000,000. The table consists of 45 pairs of facing pages. Each pair is labeled at the top with an angle, from 0 to 44 degrees, and at the bottom from 90 to 45 degrees.
The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718 281 828 459. [1] The natural logarithm of x is generally written as ln x , log e x , or sometimes, if the base e is implicit, simply log x .