Section 11.1 Rational Exponents Goals: 1. To use the properties of exponents. 2. To evaluate and simplify expressions containing rational exponents.

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1 Section 11.1 Rational Eponents Goals: 1. To use the properties of eponents.. To evaluate and simplif epressions containing rational eponents. I. Properties to Review m n A. a a = m B. ( a ) n = C. n a = b D. ( ) m ab = E. a m n a = F. G. H. I. J. m a = 1 = m a 0 a = 1 n a = m n a = II. Eamples A. Simplif: 1. ( ) ( ) B. Epress using Radicals ab 4. ( st) v Homework: p , (6 6)/, all, 71,7, 75-8 all Honors Precalculus Chapter 11 Page 1

2 Section 11. Eponential Functions Goals: 1. To evaluate epressions with irrational eponents.. To graph eponential functions.. To graph eponential inequalities. I. Graphing Eponential Functions A. The function: = a b + c 1. a represents a) b). b represents a) b). c represents = B. Graph Manuall 1. = II. Eamples 1. Suppose our parents deposited a one time amount of $5400 in an account paing 1% APR. Suppose the account paid interest monthl. Find the account balance after 18 ears.. The average growth rate of the population of a cit is 7.5% per ear and the cit s population is,750. What is the epected population in 10 ears? Homework: p all, all, 18-1 all, -9 all, 1-4 all Honors Precalculus Chapter 11 Page

3 Goals: 1. To use the eponential function Section 11. The Number e = e I. The Natural Number, e A. e = B. When it comes to the phsical world e is onl natural! J II. Some Natural Uses A. Statistics: Graph = e [-,][-1,] B. Finance: An investment compounded continuall A= Pe rt C. Science: Newton s Law of Cooling kt = ae + c D. Construction: catenar curves e = + e E. Biolog: Ebbinghaus Model ( 100 ) bt p= a e + a F. Population: P= Ce rt Homework: p ,, 4, 8-15 all, 17, 18, 0-5 all, 7, 8 Honors Precalculus Chapter 11 Page

4 Section 11.4 Logarithmic Functions Goals: 1. To evaluate epressions involving logarithms.. To solve equations involving logarithms.. To graph logarithmic functions and inequalities. I. Logarithms A. Inverse function of eponential functions B. Notation: If = a, then log a =. C. Similarities: 1 1. If = sin, then sin = = a. If =, then =. If = +, then = D. Question asked for log a b : a to what power will equal b? E. Equivalence Statements: = a loga = F. Eamples: 1. Write the following in eponential form 1 a) log8 = b) log10 = = = log. Write the folowing in logarithmic form a) 5 = b) = ( 1) 8. solve the following a) log 7( ) = b) log 8 = II. Graphing Logarithms Eamples 1. = log. = log Honors Precalculus Chapter 11 Page 4

5 III. Logarithmic Properties A. Properties: p 1. log b mn =. log m = b m. log b n = 4. log log b m = b m b or b = B. Eamples: Epand the following logarithms. 1. log. log. log 4. log C. Eamples: Write the following as a single logarithm log log log z log + log z log5 +. ( ) 1. log. z+ log 4. log 5log + log D. Eamples: Given log = and log = 1. Find: log 4. Find: log. Find: log 5. Find: log. Find: log 6. Find: 16 log 5 Honors Precalculus Chapter 11 Page 5

6 E. Solve 4log = log log (+ 5) = log (5 4). log (4+ 5) log ( ) =. 4. log ( + ) log ( 1) = log If a bacterial colon doubles ever 0 minutes and there are 500 bacteria at noon, when will the population reach 16,000? (do this two was) Homework: p. 7 1, 4, 6-16 all, 1-51 odds, 5-57 odds, 59, 6, all (eclude 70) and Worksheet Honors Precalculus Chapter 11 Page 6

7 Honors Precalculus Worksheet Section 11.4 NO CALCULATOR ON THIS WORKSHEET Name: For each of the following: Part For problems 1 8, epand each logarithm completel. Part Given log a =, log b = 7, log c =, and log d = 9, find value for each logarithm. 1. log ( ab ) b. log. log c c ab 4. log b c 5. log ( abc ) 6. log d c 7. log ab d 8. ad log cb 9. log 10. log log ( a ) 1. log 5 b a Honors Precalculus Chapter 11 Page 7

8 Write the following as a single logarithm. 1. log4 log4 z 14. loga logb+ logc 15. log ( a logc) + 5logb log5 16. log ( + ) + log ( ) log ( + 5) log ( ) 18. log ( 8+ 5) log ( ) + log ( + ) Honors Precalculus Chapter 11 Page 8

9 Section 11.5 Common Logarithms Goals: 1. To find common logarithms and antilogarithms.. To solve problems involving common logarithms. A. Common logarithms 1. The common logarithm is log 10. log 1000 =. log 100 = 4. log 0.01 = 5. log 7 = B. Converting logarithms: log b a = 1. Find log 7. Find log e C. Antilogarithms 1. Solve: = log. Solve: = log D. Applications 1. The annual growth rate of the moose population in Wells Gar Park in northern British PT log Pi Columbia, Canada, is 6.7%. This situation can be modeled b the equation n =, log1.067 where P T = population toda, P i = initial population, and n = time in ears. A few ears ago the population was 5 moose and toda it is 450 moose. Find the number of ears that have elapsed between the initial population and toda s population.. The ph of a solution is related the concentration of hdrogen ions it contains b the 1 formula ph = log, where H + is the number of gram ions of hdrogen ions per liter. If H + the ph of ocean water is 8.8, what is the concentration of hdrogen ions? Homework: p. 70 1, 8-9 all, 5-54 all, 58a, 60a, 6a, all (eclude 75) Honors Precalculus Chapter 11 Page 9

10 Section 11.6 Natural Logarithms Goals: 1. To find natural logarithms.. To solve problems involving natural logarithms. A. Natural logarithms 1. The natural logarithm is log e. ln e = 6. ln e = 4. ln1 = 5. ln = 6. ln.457 = B. Converting logarithms: log b a = 1. Find log7. Find log7 C. Solve 1. *= ln. * = ln e = 4. ln = ln e 0. Homework: p. 75 1,, 18-5 all, 6-70 all (eclude 68) Honors Precalculus Chapter 11 Page 10

11 Section 11.6/5B Solving Eponential Function Goals: 1. To solve eponential and eponential equations.. To solve eponential and eponential inequalities. Eamples 1 1. Solve two was:.9 + = 8 With common logs: With natural logs:. Phosphorus- is a radioactive substance with a half-life of 14. das. How long will it take to reduce a 100-gram sample of P- to 15 grams?. Under ideal conditions, the population of a certain bacterial colon will double in 45 minutes. How much time will it take for the population to increase five-fold? 4. Solve: 4 = 9 Honors Precalculus Chapter 11 Page 11

12 5. The eponential function = 15. ( ) models the amount of whole milk each person in the United States consumers in a ear. ( is the number of gallons of whole milk and is the number of ears since 1975.) Which ear, month, and da did whole milk consumption fall to 10.8 gal/person? 6. Solve b graphing 1 (a) = 5 (b) 1 4 < 1 Homework: p odds, 55, 57, 58b, 60b, 6b p odds, all Honors Precalculus Chapter 11 Page 1

Precalculus Chapter 10 Page 1

Precalculus Chapter 10 Page 1 Section 0. Eponential Functions. To simplify epressions and solve eponential equations involving real eponents. A. Definition of Eponential Function. An function is in the form, where and.. Graph: y =