5.6 *Exponentials, Powers, and Logarithms 289Of course, there are many possible bases to use in logarithm functions; in
fact, any positive real number other than 1 may be used. Nevertheless, two
special numbers are most often used as bases: 10 (in "common" logarithms)
and e (in "natural" logarithms). The following theorem tells us that we can
always switch from one base to another, using a conversion formula.
The orem 5 .6.25 Suppose a, b > 0, a, b =J 1, and x > 0. Then
loga x lnx 1
(a) logbx = -
1
-b = -
1
-; (b) logba = -
1
b.
oga na oga
Proof. Exercise 15. •
E XERC I SE SET 5 .6
- Prove Lemma 5.6.2.
- Prove Corollary 5.6.7.
- Prove Theorem 5.6.8 (c).
- Prove Theorem 5.6.8 (e).
- Prove Theorem 5.6.8 (f).
- Suppose a > 1. Prove that ax > 1 if x > 0, and 0 < ax < 1 if x < 0.
- Prove Theorem 5.6.12.
- Finish proving Case 1 of Theorem 5.6.14 (d), by proving (2).
- Prove Theorem 5.6. 15
- Prove that lim (1 + l)x = e.
X-+-00 X - Prove Corollary 5.6.18. [Consider one-sided limits and use Theorem 4.4.19.]
- Prove Corollary 5.6.21.
- Prove Theorem 5.6.23.
- Prove Theorem 5.6.24.
- Prove Theorem 5.6.25.
- Suppose a, b > 0, a, b =j:. 1. Prove that \:/x E JR, ax = bx logb a = ex In a.
- (Project) Prove that if a function f: JR___,(O, +oo) is strictly increasing
and \:/x, y E JR, f(x + y) = f(x)f(y), then :3 a> 1 3 \:/x E JR, f(x) =ax.
[Hint: first find j(O). Then prove that j(x) =ax if x EN, then if x E Z,
then if x E Q, and finally if x E JR.] - (Pro ject) Prove that if a function f: (0, +oo) ___.,JR is strictly increasing
and \:/x, y E JR, f(xy) = f(x)+ f(y), then :3 a> 1 3 \:/x E JR, f(x) = loga x.