PAP Algebra 2. Unit 7B. Exponentials and Logarithms Name Period

Transcription

1 PAP Algebra Unit 7B Eponentials and Logarithms Name Period

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3 PAP Algebra II Notes 7.5 Solving Eponents Same Base To solve eponential equations, get the same base on both sides of the = sign. Then the eponents must equal each other for the two sides to be equal. EX) 4 6 EX) 5 5 EX) 4 EX4) 5 8 EX5) 6 6

4 EX6) 4 EX7) EX8) EX9) ( ) 8 7

5 PAP Algebra II WS 7.5 NONCALCULATOR Name: Solve the following eponential equations for. SHOW ALL WORK! ) 5 5 ) 7 ) ) ) 5 5 6) ) 4 8) ) 9 7

6 0) ) ) ) 4) )

7 7.6 Understanding Logarithms WS Name Rewrite each equation in logarithmic form.. 4 = 8. 6 = = = = = = = = 0. 9 = 7. 6 = 6. 9 = 8 Rewrite each equation in eponential form.. log = 5 4. log 8 64 = 5. log = 6. log = 7. log 5 = 0 8. log 4 = 5 9. log 6 = log 4 = 8. log 0 = - 0. log 5 = -. log 5 8 = log = 7 Evaluate each epression 5. log log log log log log 4. log 9 7. log log 8 4. log

8 Solve each equation. 5. log b 49 = 6. log b 64 = 7. log 6 = 8. log 9 = - 9. log 6 = 40. log 7 = 4. log b 8 = 4 4. log b 8 = 4. log 5 = log = log 0 0 = 46. log 5 5 = 47. log a = log b 6 = log = log = log = log = 6 5. log 7 = 54. log 5 = log 7 = 56. log 0 0 = 57. log 6 = 58. log 6 = 6

9 PAP Algebra II 7.7A Graphing Logs & Log Properties Notes log c a b a c b EX: Write each log equation in eponential form. a) 5 log 5 b) 4 log 8 c) log(. ) d) ln EX: Find the value of each log WITHOUT A CALCULATOR. log b) a) 6 log c) log(. 00) 8 0 d) log 7 (7 ) e) ln e f) log The same transformations A, B, C and D that we used for other functions work the same way for eponential functions. y Alog b ( B( C)) D In order to graph inverses: y log we first need to graph y and use the properties of ) If the point (,y) is on f(), then (y,) is on the inverse of f(). ) The graph of a function and it s inverse are reflected across the line y=. 7

10 y log y EX: Graph the following eponential functions without a calculator. Label at least points and any asymptotes. a) y log ( ) b) y log EX4: Graph b) ylog a) log y 5 5 8

11 THREE LOG PROPERTIES:. log M log N log ( MN) addition / multiplication b M. log M log N log b b b b subtraction / divison K. M K log M log eponent b b b N EX5: Use these log properties to epand the following logs completely. a) log 7 b a b) ln 4 5 y c) log 4b d) ln y 9

12 EX6: Condense the following into a single log. Write answers in radical form. a) 5log log r 4 s q log b) log w log log z 5 c) a b) 5ln c ln d ) (ln( EX7: If log a 5 and log a 8 0 find a) log a 40 b) log a 8 c) log a 5 0

13 PAP Algebra II WS 7.7A NONCALCULATOR Name: Write each equation in eponential form. ) 6 log 6 log 4 = ) ( ) 6 = ) log 000 = 4) ln e = 5) log 8 = 4 6) ln = 0 Find the value of each log. 7) log 8) log 9 9) log 00 0) log ( ) ) ( ) 6 log ) log. 0 0 ) log ( ) 4) ( 7 9 log ) 5) log ( 0 ) ) ln e 7) ln 8) 4 lne 9) 4 ln e 0) ln ( ) ) log e 5 5 ) 7 log 6 6 ) log 9 4) log. Graph the following eponential functions. Label at least points and any asymptotes. Write domain and range for each of the functions. 5) = log 6) y = 4log( ) y

14 7) y = log + 4 8) y = log = 0) y = log ( ( ) ) 9) log y ) y = ln hint: e. 78 ) y = ln +

15 ) If f ( ) = log ( 4), find f ( ). 4) If g ( ) = + 4, find g ( ). Epand completely. 5) ln( MN ) 6) log 7 ( ) y 7) log 5 D C LM 8) log ( ) P 9) log ( a + b ) 40) 4 ln rs t

16 Condense into a single log. Write fraction eponents in radical form. 4) log log R + 4) ( ln + ln y ln z) Q 4) ( e log log log + f ) 44) ( ln p ln q) 7( ln r + ln s) 9 d ) log a log b + log c 46) 4(log( f + g) log f ) 47) Given log b = 7 and log b 6 = find a) log b b) log b c) log b 6 d) log b 6 e) log b 8 4

17 7.7B Properties of Logs Notes Pre-AP Algebra # Product Property: log b MN = log b M + log b N # Quotient Property: log b N M = log b M - log b N # Power Property: log b M K = K log b M For -5, write each epression as a single log.. log 7 + log. log 5 + log 8 log. 4 log m log n 4. log 8 log + log 5. log + log 4 log 6 For 6 0, epand. 6. log 5 s r 7. log 7( ) 8. log m 4 n - 5

18 9. log 4 (4mn) 5 0. log y For and, evaluate.. log log log - log 9. Epand log 0 in three different ways. 6

19 7.7B Properties of Logarithms Worksheet Name Epress as a sum or difference of logarithms and evaluate, if possible.. log (9 8). log (4 8). log b 7y 4. log q D log a B 6. log b 5 Epress as a product. 7. log a 4 8. log e t 6 9. log b y 0. log k 5 Epress as a single logarithm and simplify, if possible.. log + log ( 7). log a + log a y. log log y + log 4. log a ( - 9) - log a ( + ) 5. log a + log a 6. log k + log m log (n + 0) Epand. 7. log a 4y z 4 y 8. log a z 7

20 9. log a yz 0. log b 7 yz. log 5 5 y y. log z Given log = 0.0 and log 5 = 0.699, find the following without a calculator.. log 5 4. log 5. log 6. log log 5 8. log 8 8

21 Notes 7.8: Transformations of Logs PAP Algebra The General Form of a Logarithmic (Log) Equation is y = a log b ( - h) + k, where when b is not listed, it is 0. Describe the effects each of the following have on the parent function? h: k: a: First graph the parent function, then Graph y = log ( + ) Graph y = - log () + 4 Domain Range Vertical Asymptote Domain Range Vertical Asymptote 9

22 If f ( ) log0, describe the following transformations. a. g( ) log0 5 b. g( ) log 0 ( ) c. g( ) log 0 ( ) Write an equation g() for the function f ( ) log with the following transformations. A. Translated right units, up 5 units, and vertically stretched by a factor of 4. B. Reflected over the -ais, translated left units, down units, and vertically compressed by a factor of /. The function f() = log + is transformed so that the new function g() has the following points. Describe the transformation from f() to g() and write the equation of g(). a. b. c. y y y

23 Worksheet 7.8 PAP Algebra Write each equation in logarithmic form. 4 6 Write each equation in eponential form.. log log(0.00) Verbally describe the transformations of the parent function for the following:.. 5. a) y = 4 log ( ) Name: Per b) y = - 4 log () + 0 c) y = 6 log ( + ) 4 6. Graph the following using your knowledge of transformations. Then state the Domain and Range. a) f() = -log() + Transformation(s): Domain: Range: b) f() = log( + ) Transformation(s): Domain: Range: 7. Write an equation for the graph of the parent function y = log given the following transformations: a) Vertical compression by a scale factor of two-fifths. Horizontal translation right 8 units and a vertical translation down 4 units. b) Reflected over the -ais and vertically stretched by a scale factor of 6. Horizontal translation left units and a vertical translation up units. c) Horizontal translation right 7 units and vertical translation up unit.

24 8. Graph f() and g() on the same coordinate plane: y log 4 ylog 4( 4) X ¼ 4 6 Y X Y Domain: Range: Domain: Range: 9. The function f() = log + is transformed so that the new function g() has the following points. Describe the transformation from f() to g() and write the equation of g(). a. b. c. y y y Which of the following is the inverse of y 5? A. ylog 5 B. log 5 y C. ylog 5 D. ylog 5. Which of the following is the inverse of ylog 5( )? A. y 5 B. y 5 C. y 5 D. y 5. Condense: 4log (log z log y ) : Epand: log y m 6 a 4. If a and b are positive integers and 8 b, what is a b?

25 @ 6 B 5 A A : X 9 7 G : 9 A 8 a 6 ; A : < = >? ; 4 5 C < = > D E F < = >?? < = > ; G ; H I 0 J I 0 K J K J L M N O H J I J P K J Q R O L O S J I 8 T N O S J I < = >? U : V O S J I W V O S J I Z [ \ Y ] ^ _ `! " #! $ $! % & & ' & ( " ) * & +,! - - & + +! (.!.! " /

26 Notes 7.9 Applications for Logarithms Name Richter Scale of Earthquake Intensity The relationship between the magnitude of an earthquake and its intensity is logarithmic. Furthermore, the magnitude of an earthquake is related to the logarithm of the ratio of the intensity of the earthquake to the intensity of an earthquake of minimal intensity. If I 0 = is the intensity of an earthquake that can just be felt and I is the intensity of the earthquake being measured, the magnitude ( R ) of the earthquake on the Richter scale is the function R log I. I0. The largest recorded earthquake in the United States was a magnitude 9. that struck Prince William Sound, Alaska on Good Friday, March 8, 964 UTC. What was its intensity?. The largest recorded earthquake in the world was a magnitude 9.5 (Mw) in Chile on May, 960. What was its intensity?. The most recent earthquake in Chile was on Saturday Feb 7, 00 and had a magnitude of 8.8. What was its intensity? 4. How much stronger was the Chile earthquake of magnitude 9.5 then the one with magnitude of What is the magnitude of an earthquake that is 4000 times as intense as I 0? That is, I 4000I0. 6. Matt E. Matics invests $0,000 at 8% annual interest and leaves it there for 0 years. What is the total amount in his account at the end of the 0 years? 7. Stuart Dent invests $000 in an account earning 7.% compounded continuously. How many years will it take for Stu s money to double? 4

27 8. Rita Chapter invests $0,000 at 8% annual interest compounded monthly. To how much will her money nt r grow in 0 years? Use A P, where r = the annual interest rate, n = the number of compounding n periods per year, t = the time in years, P = the principal, and A = the amount in the account after t years. 9. Anita Lone invests $00,000 at 8.% annual interest compounded quarterly. After how many years will she nt r have $50,000 in her account? Use A P? n 0. Al Jebra buys a car for $9,500. The average annual depreciation on this model is %. What will the car be worth in four years when Al finally pays off his loan?. From question #5 above, when will the value of the car be half its current value?. The population of Podunk, TX is increasing at the astounding rate of 8% per year. The population is currently 64. What will the population be in 5 years?. If a paper whose thickness is.004 in is folded until it is more than. inches thick, how many times was it folded? 4. If a.000mg microorganism s size triples every week, how long until the organism has a mass of 4 mg? 5

28 PAP Algebra II Unit 7B Review Name:. Is it better to invest your money at 5.5% interest compounded continuously or at 5.8% interest compounded monthly if you have $,000 to invest for 4 years? a. If 6 b, what is a b?. If you have an account that has an interest rate of.9% compounded monthly, how long will it take for your money to triple? 4. If $000 becomes $578 with interest compounded continuously for years, what was the interest rate? 5. The model for the Richter Scale is R log I I0 where I 0. a) Find the intensity of a 6.8 quake. b) If the intensity of a quake is 5,640,000, what is the Richter scale value? c) How many times more intense is a 6. quake from an 8.7 quake? 6

29 6. Mosquitoes are tripling in number each week. If there are currently 00 mosquitoes in your bug zapper in the back yard, when will there be 000 mosquitoes? Graph the following functions. Be sure to label at least points and an asymptote. 7. y 8. y log Domain: Range: Domain: Range: 9. y 5 0. y log ( 4) Domain: Range: Domain: Range: 7

30 . If f ( ) log, find f ( ).. If the population of deer in the forest preserve is currently 00 and is growing at a rate of % per year, what is the domain for the function if the population is monitored until it reaches 000?. Epand the following completely: a) log 7 4 y z b) ln y 4 4. Condense into a single log: a) ln g ln f ln h ln( r y) 4 b) (log p 4log q log r 5log s ) 8