1. Copyright © 1999 by the McGraw-Hill Companies, Inc. Barnett/Ziegler/Byleen Precalculus: Functions & Graphs, 4 th Edition Chapter Four Exponential & Logarithmic Functions

2. 1. All graphs pass through the point (0, 1). b 0 = 1 for any permissible base b. 2. All graphs are continuous, with no holes or jumps. 3. The x axis is a horizontal asymptote. 4. If b > 1, then b x increases as x increases. 5. If 0 < b < 1, then b x decreases as x increases. 6. The function f is one-to-one. x y 1 y =b x 0 <b < 1 y =b x b > 1 DOMAIN = (– ,  )RANGE = (0,  ) Basic Properties of the Graph of f(x) = b x, b > 0, b  1 4-1-39

3. m       1 + 1 m m 12 102.59374… 1002.70481… 1,0002.71692… 10,0002.71814… 100,0002.71827… 1,000,0002.71828…...... e = 2.718 281 828 459 e π The Number e 4-2-40

4. Exponential Function with Base e 4-2-41

5. Unlimited growth Exponential decay y = ce kt c, k > 0 y = ce –kt c, k > 0 Short-term population growth (people, bacteria, etc.); growth of money at continuous compound interest Radioactive decay: light absorption in water, glass, etc.; atmospheric pressure; electric circuits Description Equation Graph Uses Exponential Growth and Decay 4-2-42-1

6. Limited growth Logistic growth y = c(1 – e –kt ) c, k > 0 Learning skills; sales fads; company growth; electric circuits Long-term population growth; epidemics; sales of new products; company growth Description Equation Graph Uses Exponential Growth and Decay y = M 1 + ce –kt c, k, M > 0 4-2-42-2

7. f xy = 2 x –3 1 8 –2 1 4 –1 1 2 01 12 24 38 f x = 2y 1 8 –3 1 4 –2 1 2 –1 10 21 42 83 Orderedpairs reversed y x y 510–5 5 10 –5 f – 1 x = 2 y or y = log 2 x f y = 2 x y =x DOMAIN off = (– ,  ) = RANGE of f –1 RANGE off = (0,  ) = DOMAIN of f –1 Logarithmic Function with Base 2 4-3-43

8. Properties of Logarithmic Functions If b, M, and N are positive real numbers, b  1, and p and x are real numbers, then: 4-3-44

9. Sound Intensity, W/m 2 Sound 1.0  10 –12 Threshold of hearing 5.2  10 –10 Whisper 3.2  10 –6 Normal conversation 8.5  10 –4 Heavy traffic 3.2  10 –3 Jackhammer 1.0  10 0 Threshold of pain 8.3  10 2 Jet plane with afterburner Sound Intensity Examples D = 10 log I I 0 Decibel scale 4-4-45

10. M = 2 3 log E E 0 Richter scale MagnitudeDestructive on Richter ScalePower M < 4.5Small 4.5 < M < 5.5Moderate 5.5 < M < 6.5Large 6.5 < M < 7.5Major 7.5 < MGreatest Earthquake Intensity Examples 4-4-46

11. Change-of-Base Formula 4-5-47