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The history of logarithms is the story of a correspondence (in modern terms, a group isomorphism) between multiplication on the positive real numbers and addition on the real number line that was formalized in seventeenth century Europe and was widely used to simplify calculation until the advent of the digital computer.
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.
Example graph of the power law, x axis represents time, y axis represents reaction time. The power law of practice states that the logarithm of the reaction time for a particular task decreases linearly with the logarithm of the number of practice trials taken. It is an example of the learning curve effect on performance.
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the 3 rd power: 1000 = 10 3 = 10 × 10 × 10.
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.
In probability theory and computer science, a log probability is simply a logarithm of a probability. [1] The use of log probabilities means representing probabilities on a logarithmic scale ( − ∞ , 0 ] {\displaystyle (-\infty ,0]} , instead of the standard [ 0 , 1 ] {\displaystyle [0,1]} unit interval .
For example, O(2 log 2 n) is not the same as O(2 ln n) because the former is equal to O(n) and the latter to O(n 0.6931...). Algorithms with running time O(n log n) are sometimes called linearithmic. [37] Some examples of algorithms with running time O(log n) or O(n log n) are: Average time quicksort and other comparison sort algorithms [38]
The worked-example effect is a learning effect predicted by cognitive load theory. [1] [full citation needed] Specifically, it refers to improved learning observed when worked examples are used as part of instruction, compared to other instructional techniques such as problem-solving [2] [page needed] and discovery learning.