Natural logarithm

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The natural logarithm is the logarithm to the base e, where e is approximately equal to 2.718... (no precise decimal fraction can be given, as e is an irrational number). The natural logarithm of x is written as ln(x). This function is the inverse function of the exponential function, thus it holds for ln(x) that eln(x) = x for all positive x and ln(ex) = x for all x.

Formally, ln(a) is defined as the the area under the graph of 1/x from 1 to a, that is,

ln(a) = 1a 1/x dx.

This defines a logarithm because it satisfies the fundamental property of a logarithm:

ln(ab) = ln(a) + ln(b).

This can be shown as follows

ln(ab) = 1ab 1/x dx = 1a 1/x dx + aab 1/x dx = 1a 1/x dx + 1b 1/x dx = ln(a) + ln(b).

The number e is then defined as the base of this logarithm.

The functions ln(x) and ex have a number of useful properties. For example, the derivative of ex is ex again (while the derivative of ax in general is ln(a).ax) and the number e itself is the limit (for n going to infinity) of (1 + 1/n)n.