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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.
The number e is a mathematical constant approximately equal to 2.71828 that is the base of the natural logarithm and exponential function.It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted .
The logarithm base 10 is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number e ≈ 2.718 as its base; its use is widespread in mathematics and physics because of its very simple derivative. The binary logarithm uses base 2 and is frequently used in computer science.
The harmonic numbers are a fundamental sequence in number theory and analysis, known for their logarithmic growth. This result leverages the fact that the sum of the inverses of integers (i.e., harmonic numbers) can be closely approximated by the natural logarithm function, plus a constant, especially when extended over large intervals.
The mathematical notation for using the common logarithm is log(x), [4] log 10 (x), [5] or sometimes Log(x) with a capital L; [a] on calculators, it is printed as "log", but mathematicians usually mean natural logarithm (logarithm with base e ≈ 2.71828) rather than common logarithm when writing "log".
Euler's formula states that, for any real number x, one has = + , where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively.
All instances of log(x) without a subscript base should be interpreted as a natural logarithm, also commonly written as ln(x) or log e (x). The first 100,000 elements of the sequence a n = log( n ) − n / π ( n ) (red line) appear to converge to a value around 1.08366 (blue line).
The base e is the most economical choice of radix β > 1, [4] where the radix economy is measured as the product of the radix and the length of the string of symbols needed to express a given range of values. A binary number uses only two different digits, but it needs a lot of digits for representing a number; base 10 writes shorter numbers ...