428 Chapter 7 • The Riemann Integral
(a) Vx E (O,oo), loga(ax) = x and a^10 g"x = x.
(b)
(c)
If a > 1, the function loga x is strictly increasing on (0, +oo) with range R
If a> 1, then lim loga x = +oo and lim loga x = -oo.
x-->+oo x-->O+
(d) log 1 ;a x = - loga x.
(e) If 0 < a < 1, loga x is strictly decreasing on (0, + oo) with range IR.
(f) If 0 <a< 1, then lim loga x = -oo and lim loga x = +oo.
x-->+oo x-->O+
(g) The "laws of logarithms" given in Theorem 7.7.3 hold for loga x.
Remarks 7.7.17 Suppose a> 0 and a -::/-1. Then
ln x
(a) loga x = -
1
-.
na
loga x
(b) Vb > 0 and b -::/-1, logb x = -
1
-b.
oga
d 1
(c) The function logax is differentiable on (O,oo), and -d logax = -
1
-.
x x na
THE ARCSINE FUNCTIONWe now direct our attention to a rigorous development of the trigonometric
functions. We would prefer to begin by defining the sine and cosine functions,
because we know t hat from them we can derive all the remaining trigonometric
functions and their interrelationships. That approach will be t aken in Chapter
8, but it requires the theory of power series. To take advantage of the Riemann
integral, we begin with the inverse sine function and use it as a foundation for
defining the sine function.^18
Definition 7.7.18 (The Arcsine Function) We define the function
arcsin x = 1x h ' for -1 < x < l.
(Recall from your calculus course why this is a reasonable definition.)
Theorem 7. 7 .19 Arcsin x has the f allowing properties on ( -1, 1):
(a) arcsin x is strictly increasing on ( -1, 1).
(b) arcsinx is continuous on (-1, 1 ).
- For an alternative development of the trigonometric functions based on the inverse tangent
fu nction, see Giaquinta and Modica [53], pp. 170 - 173.