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The natural logarithm, written mathematically as $ \ln $, is a specific case of the more familiar logarithms, denoted mathematically as $ \log $. All logarithms must have a base value. If the base is not specified, as in the case of $ \log(100) $, the base can be assumed to be ten (the common logarithm). The same holds true of the natural logarithm, which has an inherently defined base of value

$ e\approx2.71828\ldots $

(Euler's number), which is recognized by the form the function takes: $ \ln $.

  • Common Logarithm:$ \log(x)\equiv\lg(x)\equiv\log_{10}(x) $
  • Arbitrary Base Logarithm:$ \log_b(x) $
  • Natural Logarithm:$ \ln(x)\equiv\log_e(x) $

The natural logarithm $ \ln $ cannot have any other base specified. The function $ \log $, on the other hand, can have any base value other than 1 or 0. And, as already stated, if log is left without a subscripted base, its assumed base value is ten.

The natural logarithm, because of its base value $ e $, has specific properties noted in the study of calculus, which make it the ideal base value for any logarithm. Non-Euler-based logarithms do not share these properties. Some engineers, however, use their own nomenclature for the natural logarithm which is ambiguous and easily mistaken for the common logarithm:$ \log(x)\equiv\ln(x) $.

Definitions

There are various ways of giving a formal definition for the natural logarithm.

One of them is to define it is using definite integrals in the following way:

$ \ln(x)=\int\limits_1^x\dfrac{1}{u}\,du $

Another definition of the natural logarithm is the limit definition:

$ \ln(x) = \lim_{h \rightarrow 0 } \frac{x^{h} - 1 }{h} $
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