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Consider, for example, the group generated by 2 modulo = (the order of the group is =, 2 generates the group of units modulo 1019). The algorithm is implemented by the following C++ program:
Thus, log 10 (x) is related to the number of decimal digits of a positive integer x: The number of digits is the smallest integer strictly bigger than log 10 (x). [7] For example, log 10 (5986) is approximately 3.78 . The next integer above it is 4, which is the number of digits of 5986.
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. [2] [3] Parentheses are sometimes added for clarity, giving ln(x), log e (x), or log(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.
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]
Given w bits per word, the log 2 is easily computed from the clz and vice versa by log 2 (x) = w − 1 − clz(x). As demonstrated in the example above, the find first zero, count leading ones, and count trailing ones operations can be implemented by negating the input and using find first set, count leading zeros, and count trailing zeros.
Major programming languages, such as C++ (in the GNU and LLVM implementations), use introsort. [30] Quicksort also competes with merge sort, another O(n log n) sorting algorithm. Merge sort's main advantages are that it is a stable sort and has excellent worst-case performance.
A familiar example of logarithmic growth is a number, N, in positional notation, which grows as log b (N), where b is the base of the number system used, e.g. 10 for decimal arithmetic. [3] In more advanced mathematics, the partial sums of the harmonic series
Different implementations use different algorithms. The GNU Standard C++ library, for example, uses a 3-part hybrid sorting algorithm: introsort is performed first (introsort itself being a hybrid of quicksort and heap sort), to a maximum depth given by 2×log 2 n, where n is the number of elements, followed by an insertion sort on the result. [7]