Search results
Results from the WOW.Com Content Network
Logarithms and exponentials with the same base cancel each other. This is true because logarithms and exponentials are inverse operations—much like the same way multiplication and division are inverse operations, and addition and subtraction are inverse operations.
Because log(x) is the sum of the terms of the form log(1 + 2 −k) corresponding to those k for which the factor 1 + 2 −k was included in the product P, log(x) may be computed by simple addition, using a table of log(1 + 2 −k) for all k. Any base may be used for the logarithm table.
[nb 1] In some other contexts such as chemistry, however, log x can be used to denote the common (base 10) logarithm. It may also refer to the binary (base 2) logarithm in the context of computer science, particularly in the context of time complexity. Generally, the notation for the logarithm to base b of a number x is shown as log b x.
[h] The Peano axioms determine how these concepts are related to each other. All other arithmetic concepts can then be defined in terms of these primitive concepts. [147] 0 is a natural number. For every natural number, there is a successor, which is also a natural number. The successors of two different natural numbers are never identical.
Tables of common logarithms typically listed the mantissa, to four or five decimal places or more, of each number in a range, e.g. 1000 to 9999. The integer part, called the characteristic , can be computed by simply counting how many places the decimal point must be moved, so that it is just to the right of the first significant digit.
For example, log 10 10000 = 4, and log 10 0.001 = −3. These are instances of the discrete logarithm problem. Other base-10 logarithms in the real numbers are not instances of the discrete logarithm problem, because they involve non-integer exponents. For example, the equation log 10 53 = 1.724276… means that 10 1.724276… = 53.
In computer science, the iterated logarithm of , written log * (usually read "log star"), is the number of times the logarithm function must be iteratively applied before the result is less than or equal to . [1] The simplest formal definition is the result of this recurrence relation:
Unlike a linear scale where each unit of distance corresponds to the same increment, on a logarithmic scale each unit of length is a multiple of some base value raised to a power, and corresponds to the multiplication of the previous value in the scale by the base value. In common use, logarithmic scales are in base 10 (unless otherwise specified).