MATH 414
Spring 2005 Supplementary Homework #4
Due April 22
Rx
1. Define L(x) = 1 1t dt, the natural logarithm function, which we usually denote by log x or ln x. Derive the following properties using this
definition and theorems from class and the text:
(a) The domain of L is (0, ∞) and L is continuous on its domain.
(b) L0 (x) = x1 .
(c) L is strictly monotonically increasing.
(d) L(1) = 0 and there exists a unique number e > 1 such that L(e) = 1.
Rx
R xy
(e) L(xy) = L(x) + L(y) for x, y > 0. (Hint: L(xy) = 1 1t dt + x 1t dt.)
(f) L(xr ) = rL(x) for x > 0 and r ∈ Q.
(g) limx→∞ L(x) = ∞.
(h) limx→0+ L(x) = −∞.
(i) The range of L is all of R.
(j) L has the inverse function E which satisfies E 0 (x) = E(x) for all
x ∈ R.
(k) E(x) = ex for x ∈ Q
(l) If we define xα = E(αL(x)) for α ∈ R, then this agrees with the
usual definition for α ∈ Q, at least for x > 0.
(m) The derivative of xα is αxα−1 for x > 0.
L(x)
(n) lim α = 0 for any α > 0.
x→∞ x