Natural Logarithm Function

A Natural Logarithm Function is a logarithm function with Euler's number as its log function base.

• See: Base 10 Log, Exponential Function.

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References

2009

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Natural_logarithm
  • The natural logarithm, formerly known as the hyperbolic logarithm,[1] is the logarithm to the base e, where e is an irrational constant approximately equal to 2.718281828. It is also sometimes referred to as the Napierian logarithm, although the original meaning of this term is slightly different. In simple terms, the natural logarithm of a number x is the power to which e would have to be raised to equal x — for example the natural log of e itself is 1 because e1 = e, while the natural logarithm of 1 would be 0, since e0 = 1. The natural logarithm can be defined for all positive real numbers x as the area under the curve y = 1/t from 1 to x, and can also be defined for non-zero complex numbers as explained below.
  • The natural logarithm function, if considered as a real-valued function of a real variable, can also be defined as the inverse function of the exponential function, leading to the identities:
    • e^{\ln(x)} = x \qquad \mbox{if }x > 0\,\!
    • \ln(e^x) = x.\,\!
  • In other words, the logarithm function is a bijection from the set of positive real numbers to the set of all real numbers. More precisely it is an isomorphism from the group of positive real numbers under multiplication to the group of real numbers under addition. Represented as a function:
    • \ln : \mathbb{R}^+ \to \mathbb{R}.
  • Logarithms 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.

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