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) = 1∫a 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) = 1∫ab 1/x dx = 1∫a 1/x dx + a∫ab 1/x dx = 1∫a 1/x dx + 1∫b 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.