SESSION 2 Exponential and Logarithmic Functions. Math 30-1 R 3. (Revisit, Review and Revive)

Transcription

1 Mth 0-1 R (Revisit, Review nd Revive) SESSION Eponentil nd Logrithmic Functions 1

2 Eponentil nd Logrithmic Functions Key Concepts The Eponent Lws m n 1 n n m m n m n m mn m m m m mn m m m b n b b b Simplify ech epression. m. 14 bc 7 4 b c b. y 4 y 4 5y 5 y RF7. Demonstrte n understnding of rithms. 7.1 Eplin the reltionship between rithms nd eponents. 7. Epress rithmic epression s n eponentil epression nd vice vers. 7. Determine, without technoy, the ect vlue of rithm, such s Estimte the vlue of rithm, using benchmrks, nd eplin the resoning; e.g., since = nd = 4, is pproimtely equl to Notes: b 1 b nd 1 0 b Key Concepts Recll tht c c b implies tht b, nd vice vers Common Sense Rules: 0 o 1 0 since in eponentil form 1 o 1 since in eponentil form 1 o o since in eponentil form since Since our number system is bsed on powers of 10, rithms with bse 10 re widely used nd re clled common rithms. When you write common rithm, you do not need to write the bse. For emple, mens 10. If you need to work with bse other thn 10, you cn convert then to common rithms s follows: b c c b

3 Emples: 1. Rnk these rithms in order from lest to gretest: 4 6, 66, 10, 50. Evlute the following rithms using your clcultor: ) 10 c) 8 e) 6 6 g) 0.5 b) 1000 d) 5 5 f) 81 h) Evlute without clcultor ) b) 5 5 c) 5 d) 4. Convert to eponentil form nd solve where pplicble: ) y f) 5 4b 6 y j) 7 b) 5 g) 6 k) 5 7y c) 5 y l) 4 h) 9 d) 5 15 y m) 4 y e) y 851 i) y n) 5

4 5. Convert the following into rithm form: ) y d) 5y 0 g) y 5 b) y 5 10 e) y h) y 4 9 c) y f) 5 y 1 6. If 15, then find the vlue of If, 5 b, then find 5 in terms of nd b. 8. Solve If nd t, then determine the vlue of t. 4

5 10. If b 4.5 nd c.7, then the vlue of b c to the nerest tenth is The point, 8 grph of the inverse, is on the grph of the function c 1 y f. Determine the vlue of k. f nd the point (4, k) is on the 1. Convert the following rithm to the bse indicted: ) 6 16 to bse b) 00 to bse 5 5

6 RF8. Demonstrte n understnding of the product, quotient nd power lws of rithms. 8.1 Develop nd generlize the lws for rithms, using numeric emples nd eponent lws. 8. Derive ech lw of rithms. 8. Determine, using the lws of rithms, n equivlent epression for rithmic epression. 8.4 Determine, with technoy, the pproimte vlue of rithmic epression, such s 9. Notes: Chnge of bse identity cn be tught s strtegy for evluting rithms. Key Concepts: To solve those eponentil equtions when the bses re powers of the sme number, write the bses with the sme power, use the lws of eponents to simplify until you cn ignore single bse on ech side. This outcome dels with lgebric solutions to eponentil nd rithmic equtions using the lws of eponents nd rithms. The following lws of rithms re on your formul sheet, nd re useful in nswering questions in this outcome. M N M N M M N n M n M N Emples: 1. Solve: ) d) g) 5 45 b) e) 1 8 h) 16 1 c) f) 1 6

7 . Evlute: 1 ) e) b) f) 5 c) g) d) 6 8. Write the following s single rithm. ) B D 5 E A A 1 b) 4 c) 46 y 6 y 6 y 4. Epnd ech epression using the lws of rithms. ) 4 y 4 z b) 5 y 7

8 c) 100 y 4 5. If A t then 7A 6. If y c, then y is equl to 7. Solve for. ) 4 1 b) 1 7 c) Solve for. 4 ) c) b b b b) d) 7 5 8

9 e) g) f) h) y y 9

10 RF9. Grph nd nlyze eponentil nd rithmic functions. [C, CN, T, V] [ICT: C6 4., C6 4.4, F1 4.] 9.1 Sketch, with or without technoy, grph of n eponentil function of the form y =, > Identify the chrcteristics of the grph of n eponentil function of the form y =, > 0, including the domin, rnge, horizontl symptote nd intercepts, nd eplin the significnce of the horizontl symptote. 9. Sketch the grph of n eponentil function by pplying set of trnsformtions to the grph of y =, > 0, nd stte the chrcteristics of the grph. 9.4 Sketch, with or without technoy, the grph of rithmic function of the formy = b, b > Identify the chrcteristics of the grph of rithmic function of the form y = b, b > 1, including the domin, rnge, verticl symptote nd intercepts, nd eplin the significnce of the verticl symptote. 9.6 Sketch the grph of rithmic function by pplying set of trnsformtions to the grph of y = b, b > 1, nd stte the chrcteristics of the grph. 9.7 Demonstrte, grphiclly, tht rithmic function nd n eponentil function with the sme bse re inverses of ech other. Notes: - When grphing y b nd y, the vlue of b will be restricted to b0, b 1 b - Nturl rithms nd bse e re beyond the scope of this course. Key Concepts An eponentil function is written in the form y b, where b is constnt nd is the eponent. It is incresing when b > 1, nd decresing when 0 < b < 1. Domin: Rnge: -intercept: y-intercept: Horizontl Asymptote: Domin: Rnge: -intercept: y-intercept: Horizontl Asymptote: A rithmic function is the inverse of n eponentil function. Remember, the line y = cts s mirror, to find the inverse of function, we tke the following steps: Step #1 switch nd y Step # solve for y But for the inverse of n eponentil function it is difficult to solve for y. In the inverse of y y the first step is to switch nd y, therefore, but it is difficult to solve for y. To solve for y we introduce the rithmic function s follows: y y Log form Eponentil form 10

11 Eponentil Form Domin: Rnge: -intercept: y-intercept: Asymptote: Log Form Domin: Rnge: -intercept: y-intercept: Asymptote: Emples: 1. Stte which of the following re eponentil. ) y 7 c) y 4 e) y g) 1 y b) y 0.5 d) 1 y f) y. Determine the inverse of ech: ) y b) y 4 c) 0 y. Stte the trnsformtions, symptotes nd mpping nottion for ech, compred to y nd y Asymptote ) y 5, y 11

12 Asymptote 1 b) y 6 y, y 4 c) y, 1 5 d) 1 y 5, y 4. Describe the trnsformtion which mps the grph of y 9 to the grph of the following: ) 1 b)

13 5. Write the eqution of the trnsformed rithmic function for y 5 for ech of the following: ) the verticl stretch bout the -is by fctor of 1 nd horizontl stretch bout the y-is by fctor of nd reflected in the -is nd horizontl trnsltion of units to the left. b) A verticl stretch by fctor of bout the -is, horizontl stretch by fctor of 1 bout the y-is, 4 reflected on the y-is, horizontl trnsltion 1 to the right nd verticl trnsltion of 5 units down. 6. The grph of y f b, where b > 1, is trnslted such tht the eqution of the new grph is epressed s y f 1. Identify the rnge nd the y-intercept of the new function. 7. The eqution of the symptote of y Stte the domin, rnge, nd y intercepts nd symptotes of y 9. Identify the eqution of the symptote of y 1 1 1

14 RF10. Solve problems tht involve eponentil nd rithmic equtions. [C, CN, PS, R] 10.1 Determine the solution of n eponentil eqution in which the bses re powers of one nother. 10. Determine the solution of n eponentil eqution in which the bses re not powers of one nother, using vriety of strtegies. 10. Determine the solution of rithmic eqution, nd verify the solution Eplin why vlue obtined in solving rithmic eqution my be etrneous Solve problem tht involves eponentil growth or decy Solve problem tht involves the ppliction of eponentil equtions to lons, mortgges nd investments Solve problem tht involves rithmic scles, such s the Richter scle nd the ph scle Solve problem by modelling sitution with n eponentil or rithmic eqution. Notes: - Formuls will be given for ll problems involving rithmic scles such s decibels, erthquke intensity, nd ph p - Formuls will be given unless the contet fits the form y b, where y is the finl mount, is the initil mount, b is the growth/decy fctor, t is the totl time, nd p is the period time. - Logrithmic equtions should be restricted to the sme bses. Key Concept: To solve these eponentil equtions when the bses re powers of the sme number, write the bses with the sme power, use the lws of eponents to simplify until you cn ignore single bse on ech side. This outcome dels with lgebric solutions to eponentil nd rithmic equtions using the lws of eponents nd rithms. The following lws of rithms re on your formul sheet, nd re useful in nswering questions in this outcome. M N M N M M N The compound interest eqution, A P1 i of popultion. n M n M n 14 t N, cn be used to model the growth or decy of ny type For pplictions with interest A is the mount of money t the end of the investment, the finl mount P is the principl mount deposited, the initil mount deposited i is the interest rte per compounding period, epressed s deciml (interest rte / # compounding periods) n is the number of compounding periods, the number of times you get interest (# yers # compounding periods) i n Annully 1 1 Semi-nnully Qurterly 4 4 Monthly 1 1

15 For ll other word problems, deling with Growth or Decy, the following formul will be used y the finl mount is the initil mount t y b p b is the multipliction fctor: for doubles, ½ for hlf-life, for triples t time, must be in the sme units s p p the period of time for doubling, hlf-life, or tripling. Emples: 1. The popultion of city ws on Jnury 1, 1988 nd it ws on Jnury 1, 00. If the growth of the city cn be modeled s n eponentil function, then find the verge nnul growth rte of the city, epressed to the nerest tenth of percent.. A sports cr deprecites t rte of 14% per yer nd ws bought for $ How long will it tke to deprecite to $18 000?. Write n eponentil epression tht will determine the vlue, V, of the investment t ny given time, t, in yers. ) $000 is invested t 5.% per yer compounded semi-nnully b) $500 is invested t 4% per yer compounded qurterly c) $8000 is invested t 6% per nnum compounded monthly 4. A student borrowed $6000. Interest is chrged t 5% per yer compounded semi-nnully. The lon is pid off in one finl pyment of $6958. Wht is the length of the lon? 15

16 5. Jmie borrows $6000 from the bnk t rte of 8% per nnum compounded monthly. How much would he owe t the end of the one month, if he does not mke his first pyment? 6. The popultion of rbbits in prk is incresing by 70% every 6 months. Presently there re 00 rbbits in the prk. ) Write n eponentil function tht represents this scenrio. Use P to represent the rbbit popultion nd t to represent the time in months. b) How mny rbbits will there be in 15 months? 7. A smple of wter contins 00 grms of pollutnts. Ech time the smple is pssed through filter, 0% of its pollutnts re removed. ) Write function tht reltes the mount of pollutnt, P, tht remins in the sme to the number of times, t, the smple is filtered. b) Determine n epression tht gives the mount of pollutnts still in the wter fter it psses through 5 filters. How mny grms re there fter 5 filters, rounded to the tenth of grm? 8. A colony of 100 insects triples its popultion every 5 dys. How long will it tke for the popultion to grow to 5000? 9. A type of bcterium doubles ech hour. ) If there re 4 bcteri in smple, write n eponentil function tht models the smple s growth over time. b) Use your eqution to determine the time it tkes for the smple to become 4096 bcteri. 16

17 10. The hlf-life of rdioctive 14 is 8.1 dys. Determine the number of dys it took for there to be only % remining? 11. The sound intensity, B, in decibels, is defined s B 10 I, where I is the intensity of the sound. A I fine cn occur when motorcycle is idling t 0 times s intense s the sound of n utomobile. If the level of n utomobile is 80 db, t wht decibel level cn fine occur to motorcycle opertor? 0 1. A sound is 1000 times more intense thn sound you cn just her. Wht is the mesure of its loudness in decibels? 1. How mny more times intense is the sound of norml converstion (60 db) thn the sound of whisper (0 db)? 14. The ph scle is used to mesure cidity or lklinity of substnce. The formul for ph is ph H, where [H+ ] is the hydrogen ion concentrtion. ) If solution hs hydrogen ion concentrtion of ml/l, determine the ph vlue of the solution to the nerest tenth. b) If vinegr hs ph of.. Determine the [H + ] in scientific nottion to one deciml plce. 17

18 15. Erthquke intensity is given by I I 10 m, 0 where I 0 is the reference intensity nd m is the mgnitude. An erthquke with mgnitude 6.8 is followed by n ftershock with mgnitude 5.. How mny time more intense ws the erthquke thn its ftershock? 16. The formul for the Ritcher mgnitude Scle of n erthquke is M I, where I is the I mgnitude of the lrgest erthquke nd I 0 is the mgnitude of the smllest erthquke. How much more intense is n erthquke with mgnitude of 6.9 vs.9? Erthquke intensity is given by I I 10 m, 0 where I 0 is the reference intensity nd m is the mgnitude. A prticulr mjor erthquke of mgnitude is 10 times s intense s prticulr minor erthquke. The mgnitude, to the nerest tenth, of the minor erthquke is. Prctice Test 1. The solution for the eqution 9 A. 6 B. 4 C. 6 D is. In rithmic form, the solution of 50 5 A B C D is 18

19 . The prtil grph of the eponentil function f 4 is shown to the right. The domin of the inverse function, f 1 is A. 0, B. 0, C. 0, D. 4. The vlue of in the eqution A B..97 C..97 D is 5 1, then the vlue of is 5. If 8 8 A. - B. 1 C. D. 8 19

20 6. r t d is equl to t t A. rd B. rd C. r d t D. r d 4 7. The epression b b A. B. 4 C. D. is equivlent to 8. A student grphed the following equtions. Eqution I y1 10 Eqution II y 5 Eqution III y Eqution IV y4 The student could estimte the solution to the eqution 5 by using the grphs of equtions A. I nd II B. I nd III C. II nd III D. II nd IV t N t N 0, where N(t) = the finl number of bcteri, N 0 = the initil number of bcteri, nd t = time in minutes. The rithmic epression for time (t) it tkes for the number of bcteri to increse from to is (SE) 40 A The growth of bcteri cn be written 40 B C t D. 40 0

21 10. The vlue of i in the compound interest eqution 51 i 6 is A. B. C. D. i 6 4 i i i The gin, G, mesured in decibels, of n mplifier is defined by the eqution G 10 P where P P1 is the output power of the mplifier, in wtts, nd P1 is the input power of the mplifier, in wtts. If the P gin of the mplifier is decibels, then the rtio P is A. 10 B.. C. D I 1. The eqution tht defines the decibel level for ny sound is L 1010, where L is the loudness I0 in decibels, I is the intensity of sound being mesured, nd I0 is the intensity of sound t the threshold of hering. Given tht norml converstion is times intense s I0, then the loudness of norml converstion is A. 5 decibels B. 6 decibels C. 16 decibels D. 60 decibels 1. A restriction on the domin of A. 4 B. 0 C. D. 4 1 f 4 such tht its inverse is lso function, could be (SE) 1

22 14. The grph of y is reflected in the line y =. The eqution of the trnsformed grph is A. y B. y C. y D. y 15. The eqution m p n q cn be written in eponentil form s A. B. C. D. m p q m p q p qm p nq n mn n 16. The epression is equivlent to A. B. C. D Written s single rithm, A. y z B. y z C. y z D. z y z y is

23 Use the following informtion to nswer the net question. A student s work to simplify rithmic epression is shown below, where > 1. 4 STEP 1 4 STEP STEP STEP 4 STEP The student mde his first error when going from: A. Step 1 to Step B. Step to Step C. Step to Step 4 D. Step 4 to Step The eqution of the symptote for the grph of y A. = B. = - C. = D. = -, where b > 0 nd b 1, is b 0. The grph of y is trnsformed into the grph of y 4 5 i nd 4 units ii. The sttement bove is completed by the informtion in row Row i ii A. right up B. left up C. right down D. left down by trnsltion of 5 units

24 1. For the grph of y 1, where 0 < b < 1, the domin is b A. > -4 B. > 4 C. > -1 D. > 1. The y-intercepts on the grph of 1 f b is A. B. b C. 1 + b D. + b Numericl Response 1. If.6 nd b.7 then n n n b correct to the nerest tenth is.. If b = 4, then to the nerest hundredth, the vlue of b, where b, 0, is.. The inverse of f is written in the form,,,. 1 c f, the vlues of, b, c, d re b d 1 4. The vlue of bb 1 is Given tht 6 nd 5, b determine the vlue of b Algebriclly solve the eqution

25 7. Solve the eqution lgebriclly. Round to the nerest hundredth, if necessry Wht is the etrneous root? (SE) 8. Solve lgebriclly 7 7 Use the following informtion to nswer the net question. M Erthquke intensity is given by I I0 10, where I 0 is the reference intensity nd M is the mgnitude. An erthquke mesuring 5. on the Richter scle is 15 times more intense thn second erthquke. 9. Determine, to the nerest tenth, the Richter scle mesure of the second erthquke. 10. The popultion of prticulr town on July 1, 011 ws If the popultion decreses t n verge nnul rte of 1.4%, how long will it tke for the popultion to rech 15 00? (SE) 5

26 Eponents nd Logrithm Prctice Test Answers: MULTIPLE CHOICE: 1. D. C. A 4. C 5. C 6. B 7. A 8. D 9. C 10. B 11. D 1. D 1. B 14. A 15. B 16. A 17. A 18. B 19. C 0. B 1. A. D NUMERICAL RESPONSE: ,,,