The definition
Like ex (see also exp), ln(x) (read "the natural logarithm of x") is a function defined for the purposes of making some calculus questions easier. The natural logarithm is defined to be "the function whose derivative is 1/x". In symbolic form:
d 1
-- (ln(x)) = -
dx x
This is equivalent to the definition given in
ln.
Properties
All the usual properties of logarithms hold for the natural logarithm, for example:
log x
28
ln x = log x = ------
e log e
28
(where 28 is an arbitrary real number)
a
ln (x ) = a ln x
So far, the reasons why ln(x) is a logarithmic function and its base is e escape me. Maybe someday I'll understand enough analysis to present them here. For further reading, check out http://mathworld.wolfram.com/NaturalLogarithm.html.