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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.
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The full formula, together with precise estimates of its error, can be derived as follows. Instead of approximating !, one considers its natural logarithm, as this is a slowly varying function: (!) = + + + .
If p is a probability, then p/(1 − p) is the corresponding odds; the logit of the probability is the logarithm of the odds, i.e.: = = = = (). The base of the logarithm function used is of little importance in the present article, as long as it is greater than 1, but the natural logarithm with base e is the one most often used.
Logarithmic number systems have been independently invented and published at least three times as an alternative to fixed-point and floating-point number systems. [1]Nicholas Kingsbury and Peter Rayner introduced "logarithmic arithmetic" for digital signal processing (DSP) in 1971.
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 Chebyshev function ψ ( x ) {\displaystyle \psi (x)} , with x < 50 The function ψ ( x ) − x {\displaystyle \psi (x)-x} , for x < 10 4 The function ψ ( x ) − x {\displaystyle \psi (x)-x} , for x < 10 7
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