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 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) .


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

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


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