Figure N6–16Note that if x = 1 the above definition yields ln 1 = 0, and if 0 < x < 1 we can
rewrite as follows:
,showing that ln x < 0 if 0 < x < 1.
With this definition of ln x we can approximate ln x using rectangles or
trapezoids.
Example 35 __
Show that .
SOLUTION: Using the definition of ln x above yields , which we
interpret as the area under , above the t-axis, and bounded at the left by t = 1
and at the right by t = 2 (the shaded region in Figure N6–16). Since is
strictly decreasing, the area of the inscribed rectangle (height , width 1) is less
than ln 2, which, in turn, is less than the area of the circumscribed rectangle
(height 1, width 1). Thus
.