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.
/Talk