文档介绍：Section
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The Natural Logarithm Function
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In the last example of Section we saw the need for solving an equation of the form
ex = b.
for x in terms of b. In general, for a given function f, a function g defined on the range of
f is called the inverse of f if
g(f(x)) = x ()
for all x in the domain of f and
f(g(x)) = x ()
for all x in the domain of g. That is, if f(x) = y, then g(y) = x and if g(x) = y, then
f(y) = x. In order for a function f to have an inverse function g, for every point y in the
range of f there must exist a unique point x in the domain of f such that f(x) = y, in
which case g(y) = x. In other words, for any two points x1 and x2 in the domain of f, we
must have f(x1) =6 f(x2). Now this will be the case if f is increasing on its domain, since,
for such an f, x1 < x2 implies f(x1) < f(x2). In particular, a function f with domain
(a, b) will have an inverse if f 0(x) > 0 for all x in (a, b). Hence, since
d
ex = ex > 0
dx
for all x in (−∞, ∞), the function f(x) = ex must have an inverse defined for every point
in its range, namely, (0, ∞). We call this inverse function the natural logarithm function.
Definition The inverse of the exponential function is called the natural logarithm func-
tion. The value of the natural logarithm functi