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Section 3.2 Logarithmic Functions and Their Graphs 205
Section 3.2
Logarithmic Functions and Their Graphs
■ You should know that a function of the form y log a x, where a > 0, a 1, and x > 0, is called a
logarithm of x to base a.
■ You should be able to convert from logarithmic form to exponential form and vice versa.
y log a x ⇔ a y x
■ You should know the following properties of logarithms.
(a) log a 1 0 since a 0 1. (c) log a a x x since a x a x .
(b) log a a 1 since a 1 a. (d) If log a x log a y, then x y.
log e x ln x, x > 0
■ You should know the definition of the natural logarithmic function.
■ You should know the properties of the natural logarithmic function.
(a) ln 1 0 since e 0 1. (c) ln e x x since e x e x .
(b) ln e 1 since e 1 e. (d) If ln x ln y, then x y.
■ You should be able to graph logarithmic functions.
Vocabulary Check
1. logarithmic function 2. 10 3. natural logarithmic
4. a log a x x
5. x y
1. log 4 64 3 ⇒ 4 3 64 2. log 3. log 1 7 49 2 ⇒ 72 1 3 81 4 ⇒ 3 4 81
49
1
4. log 5. log 32 4 2 5 ⇒ 10 1000 3 ⇒ 103 1
1000 3225 4 6. log 16 8 3 4 ⇒ 1634 8
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7. ln 1 0 ⇒ e 0 1
8. ln 4 1.3862 . . . ⇒
9. ln e 1 ⇒ e 1 e
e 1.3862 . . . 4
10. ln e 3 3 ⇒ e 3 e 3 11. ln e 1 2 ⇒ e12 e 12. ln 1 e 2 ⇒ 2 e2 1 e 2
13. 5 3 125 ⇒ log 5 125 3 14. 8 2 64 ⇒ log 8 64 2 15. 81 14 3 ⇒ log 81 3 1 4
16. 9 32 27 ⇒ log 9 27 3 2
18. 10 3 0.001 ⇒ log 10 0.001 3
17. 6 2 1 36 ⇒ log 1 6 36 2
19. e 3 20.0855 . . . ⇒ ln 20.0855 . . . 3
20. e 4 54.5981 . . . ⇒ ln 54.5981 . . . 4 21. e 1.3 3.6692 . . . ⇒ ln 3.6692 . . . 1.3
22. e 2.5 12.1824 . . . ⇒ ln 12.1824 . . . 2.5 23. 3 e 1.3956 . . . ⇒ ln1.3956 . . . 1 3