F. INTERPRETING ln x AS AN AREA
It is quite common to define ln x, the natural logarithm of x, as a definite integral, as follows:
This integral can be interpreted as the area bounded above by the curve below by the t-
axis, at the left by t = 1, and at the right by t = x (x > 1). See Figure N6–16.
FIGURE N6–16
Note 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 < ln 2 < 1.
SOLUTION: Using the definition of ln x above yields which we interpret as the area