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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 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.
When the base is clear from the context or is irrelevant it is sometimes written log x. 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.
With base e the natural logarithm behaves like the common logarithm as ln(1 e) = 0, ln(10 e) = 1, ln(100 e) = 2 and ln(1000 e) = 3. 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.
The "natural" base = is the unique base satisfying the criterion that the exponential function's derivative equals its value, = , which simplifies definitions and eliminates extraneous constants when using exponential functions in calculus.
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.
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is Euler's number, the base of natural logarithms, is the imaginary unit, which by definition satisfies =, and is pi, the ratio of the circumference of a circle to its diameter. Euler's identity is named after the Swiss mathematician Leonhard Euler.