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Graph of the natural logarithm function. The function slowly grows to positive infinity as x increases and slowly goes to negative infinity as x approaches 0 ("slowly" as compared to any power law of x).

The natural logarithm is the logarithm to the base e, where e is an irrational constant approximately equal to 2.718281828. In simple terms, the natural logarithm of a number x is the power to which e would have to be raised to equal x. It is in other words the inverse function of the exponential function, leading to the identities:

$e^{\ln(x)}=x,\qquad {\mbox{if }}x>0\,\!$

$\ln(e^{x})=x\,\!$

The natural logarithm of x can also be defined as the area under the curve y = 1/t, between t = 1 and t = x, for all positive real numbers x.

Logarithms in general can be defined to any positive base other than 1, not just e, and are useful for solving equations in which the unknown appears as the exponent of some other quantity.

History

The first mention of the natural logarithm was by Nicholas Mercator in his work Logarithmotechnia published year 1668, altough the mathematics teacher John Speidell had already in 1619 compiled a table on the natural logarithm. It was formerly also called hyperbolic logarithm.