Precalculus Chapter 10 Page 1

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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 = 5 a) What is the domain and range of the function? b) Any asymptotes? B. Review. General Equation: y = Ca. Initial Value. Growth vs. Decay a) Growth () () b) Decay () () What is the eponential function whose graph passes through the points (0,-5) and (-,-5)? C. Simplify Epressions (Eamples) ( 4 ) 4. ( ) 5 y D. Solving Eponential Equations in the form a = a. Theorem:. Eamples a) 4 = 7 m b) 6 n+ = c) n 5 = 5 n 5 Homework p all,, 5, (7-54)/, 55, all, 6-64 all, all Precalculus Chapter 0 Page

2 Section 0. Logarithms and Logarithmic Functions. To evaluate epressions involving logarithms.. To solve equations involving logarithms.. To graph logarithmic functions and inequalities. I. Logarithms A. function of eponential functions B. Notation: If y C. Similarities: = a, then log a y. If y =, then y = =. y = a y = y = log. If y =, then y =. If y = +, then y = D. Question asked for by the symbol log a b : to what will equal? E. Equivalence Statements: y = a log a y = Similar to.... y = y =. y = y =. y = + y = F. Eamples:. Write the following in eponential form a) log8 = b) log0 =. Write the folowing in logarithmic form y a) 5 = b) = ( ) 8 y. Solve the following a) log 7( ) = b) log 8 = c) logb 5 = d) log ( + 5) = log (5 4) Precalculus Chapter 0 Page

3 . Simplify a) 8 log5 5 b) log ( ) II. Graphing Logarithms Eamples. y = log. y = log Homework: p. 55, - all, -59 odds, all,74, 78-8 all Precalculus Chapter 0 Page

4 Section 0. Properties of Logarithms. To simplify and evaluate epressions using properties of logarithms. To solve equations involving logarithms I. Logarithmic Properties A. Properties:. log b mn =. log m p = b m. log b n = **4. log log b m = b m b or b = B. Eamples: Given log =.5850 and log 5 =.9 5. Find: log 8 4. Find: log 9 4. Find: log 5. Find: log 64. Find: log 6 6. Find: log 5 C. Solve. log (4 + 5) log ( ) =. log ( + ) log ( ) = log log8 = log8 8 Homework: p. 544, - odds, 8, 9, 44, 45, 49, 5-6 odds Precalculus Chapter 0 Page 4

5 Section 0.4 and Section 0.5 Common Logarithms and Natural Logarithms. To find common logarithms and antilogarithms.. To find natural logarithms and antilogarithms.. To solve problems involving common logarithms. I. Common Logarithms A. The common logarithm(log) is log 0 4. log 000 = 5. log 00 = 6. log 0.0 = 7. log 7 = D. Converting logarithms: log b a =. Find log 7. Find loge E. Antilogarithms. Solve: = log. Solve: = log. The ph of a solution is related the concentration of hydrogen ions it contains by the formula ph = log, where H + is the number moles per liter of hydrogen ions. If the H + ph of ocean water is 8.8, what is the concentration of hydrogen ions? II. Natural Logarithms A. The natural logarithm(ln) is log e. ln e. ln e 5. ln 7 B. Antiln. = ln = ln e = 4. ln = ln e A major highway was constructed five years ago to accommodate a population of up to 40,000 commuters. It is estimated that the commuter population at that time was about 5,000. Today, there are,000 cars commuting on the highway each day. If the commuter population continues to grow at this rate, when will the highway need to be upgraded again? Homework: p odds, 45-5 odds, 5-55 all, 60 and p. 6, -7 odds, 47-5 odds, 58, 60, Precalculus Chapter 0 Page 5

6 Section 0.4B and Section 0.5B Solving Eponential and Logarithmic Equations. To solve eponential and eponential equations. Eamples. Solve two ways:.9 + = 8 With common logs: With natural logs:. Phosphorus- is a radioactive substance with a half-life of 4. days. How long will it take to reduce a 00-gram sample of P- to 5 grams?. Under ideal conditions, the population of a certain bacterial colony will double in 45 minutes. How much time will it take for the population to increase five-fold? 4. Solve: 4 = 9 5. The eponential function y = 5. ( ) models the amount of whole milk each person in the United States consumers in a year. (y is the number of gallons of whole milk and is the number of years since 975.) Which year, month, and day did whole milk consumption fall to 0.8 gal/person? Homework: p odds, 57, 58, 6, and p. 557, 9-45 odds,58, 6 Precalculus Chapter 0 Page 6

7 Section 0.6 Growth and Decay Reviewed and Epanded Goal:. To use logarithms or natural logarithms to solve problems involving growth and decay. Note: Assume continuous growth or decay unless told otherwise. Formulas: rt Continuous: y = Ce r Not Continuous: y = C + n nt Eamples:. For a certain radioactive element, the rate of decay is 0.77 when t is measured in days. How long will it take 500 grams of the element to reduce to 00 grams?. The Jamisons bought a new house 5 years ago for $65,000. The house is now worth $7,000. Assuming a continuous rate of growth, what was the yearly rate of appreciation?. The Thomas family includes several generations of farmers. They have an opportunity to buy 50 acres adjacent to their farm for $800 per acre. In the past, the price of farmland has gone up % per year. If this continues, how long will it be before the land is worth $000 per acre? Homework p , 0,, 4 Precalculus Chapter 0 Page 7