Natural logarithm

Natural logarithm
Representation	${\displaystyle \ln x}$
Inverse	${\displaystyle e^{x}}$
Derivative	${\displaystyle {\frac {1}{x}}}$
Indefinite Integral	${\displaystyle x\ln x-x+C}$

The natural logarithm of a number is its logarithm to the base of the mathematical constant e, where e is an irrational and transcendental number approximately equal to 7000271828182845899♠2.718281828459. The natural logarithm of x is generally written as ln x, log_ex, or sometimes, if the base e is implicit, simply log x.Parentheses are sometimes added for clarity, giving ln(x), log_e(x) or log(x). This is done in particular when the argument to the logarithm is not a single symbol, to prevent ambiguity.

The natural logarithm of x is the power to which e would have to be raised to equal x. For example, ln(7.5) is 2.0149..., because e^2.0149... = 7.5. The natural log of e itself, ln(e), is 1, because e¹ = e, while the natural logarithm of 1, ln(1), is 0, since e⁰ = 1.

The natural logarithm can be defined for any positive real number a as the area under the curve y = 1/x from 1 to a (the area being taken as negative when a < 1). The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural". The definition of the natural logarithm can be extended to give logarithm values for negative numbers and for all non-zero complex numbers, although this leads to a multi-valued function: see Complex logarithm.

The natural logarithm function, if considered as a real-valued function of a real variable, is the inverse function of the exponential function, leading to the identities:

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