Math 409 Homework 6, due Thursday March 23, 2017.
Let x > 0 be a nonnegative real number and let k ∈ N. We proved
1
in class that there is a unique real number, denoted x k , satisfying (i)
1
1
x k > 0 and (ii) (x k )k = x.
Definition 1: For x > 0 and k ∈ N, we define
x−k :=
1
1
1
and x− k := 1 .
k
x
xk
Definition 2: For x > 0 and a rational number q ∈ Q define
1
xq := (x m )n
where n, m ∈ Z are such that q =
n
.
m
1. (a.) Prove that xq is well-defined, i.e., suppose that
1
m0
n0
m0
=
1
m1
n1
,
m1
n1
where n0 , n1 , m0 , m1 ∈ Z. Prove that (x )n0 = (x )
for all x > 0.
(b.) Let p, q ∈ Q. Prove that (xp )q = xpq , and that xp xq = xp+q
for all x > 0.
(c.) Suppose x > 1 and that p and q are rational numbers with
p < q. Prove that xp < xq .
2. Let x be a real number with x > 1. For each r ∈ R define
fr (x) := sup{xq : q ∈ Q and q ≤ r}.
(a.) Prove that fr (x) is finite and that fr (x) = xr whenever
r ∈ Q. Thus, the definition xr := fr (x) for x > 1 and
r ∈ R is consistent with our previous definition of xr for
r ∈ Q give in problem 1 above.
(b.) Assume r, s ∈ R and r < s. Prove that xr < xs .
(c.) Let g : R → R be the function defined by g(r) := xr . Prove
that g is continuous.
(d.) Let r, s ∈ R. Prove that (xr )s = xrs , xr xs = xr+s , and
x−r = 1/xr .
(e.) Let 0 < y < 1 and define y r := (1/y)−r . Prove that (c.)
and (d.) hold for y in place of x.
3. (a.) Let k ∈ Z and define f (x) = xk . Prove that f is differentiable at every x ∈ (0, ∞) and that f 0 (x) = kxk−1 . (In
class we showed this is true for k ∈ N. This problem asks
you to show that it is true for all k ∈ Z.)
1
2
1
(b.) Let k ∈ Z and define g(x) = x k . Prove that g is differen1
tiable at every x ∈ (0, ∞) and that g 0 (x) = k1 x k −1 .
(c.) Let q ∈ Q and let h(x) = xq . Prove that h is differentiable
at every x ∈ (0, ∞) and that h0 (x) = qxq−1 .
4. Let y > 1 be a real number and let a > 0.
(a.) Prove that there exists a unique x ∈ R such that y x = a.
We denote this unique real number x by x = logy (a). Thus,
logy : (0, ∞) → R is the unique function which satisfies y logy (a) =
a for every real number a.
(b.) Prove that logy (ab) = logy (a)+logy (b), and that logy (ar ) =
r logy (a) for any a, b > 0 and r ∈ R.
(c.) Prove that if 0 < a < b then logy (a) < logy (b).
(d.) Prove that logy is continuous on (0, ∞).