Exponentials & Logarithms Unit 8

Transcription

1 U n i t 8 AdvF Dte: Nme: Eponentils & Logrithms Unit 8 Tenttive TEST dte Big ide/lerning Gols This unit begins with the review of eponent lws, solving eponentil equtions (by mtching bses method nd tril & error method) nd problems solving with eponentil functions. You will then lern bout the inverse of eponentil functions which is logrithmic function of the sme bse. Just like there re severl eponent lws there will be severl logrithmic lws tht you will hve to know. These lws will help you in solving more complicted eponentil equtions where previous methods don t work or to void the use of tril & error method. The lws will lso help you solve logrithmic equtions. You will lern how to grph logrithmic functions by the use of trnsformtions or by the use of key chrcteristics nd finlly you will study some rel life situtions tht involve logrithms. Corrections for the tetbook nswers: Sec 8. #9c). Sec 8. #4d) iii) domin >0 #8) (5, -) Sec 8. #4d).40 Sec 8.4 #0c) =4 Sec 8.5 #6) 9.0 Sec 8.6 #0 = the only solution #4) mph Sec 8.8 #7 tble is wrong for it to be eponentil with sme growth rte Review #7d) log44 Success Criteri I understnd the new topics for this unit if I cn do the prctice questions in the tetbook/hndouts Dte pg Topics # of quest. done? You my be sked to show them -4 Review of Grde ( dys if there s time) THREE Hndouts Eponent lws, Solve eponentil equtions by mtching bses, Solve eponentil word problems by tril & error 5-6 Wht is Logrithmic Function? Section 8. & 8. & THREE Hndouts 7-8 Eponentil & Logrithmic Functions Section 8. & THREE Hndouts 9-0 Lws of Logrithms Section 8.4 & TWO Hndouts - Solving Eponentil Equtions by using Logs Section 8.5 & TWO Hndouts -4 Solve Logrithmic Equtions Section 8.6 & TWO Hndouts 5-6 Solve Problems Section 8.7 & Hndout REVIEW Questions I hd difficulty with sk techer before test! Reflect previous TEST mrk, Overll mrk now.

2 U n i t 8 AdvF Dte: Nme: Review of Grde. Summrize the lws you lerned in grde 9- (multipliction, division, power of power, zero, negtive, rtionl, distributive properties). Apply the lws to the following emples s you simplify the questions. Leve everything s ect numbers, with positive eponent nswers.. ( y y ) 6 y b. ( 8 ) ( 7 ) y y c. 64m 4 5 d. c 4 ( c ) ( c ) e. 8 y 4 f. ( ) 4 4

3 U n i t 8 AdvF Dte: Nme: There re severl useful constnts tht re used for mth π is clled nd π = is used with nything circulr e is clled nd e = is used with eponentil continuous growth/decy Red up more on the number e: When the vrible is on the bse, not in the eponent, to solve it you must isolte it by using When the vrible is in the eponent, not on the bse, to solve it you must When the vrible is in the eponent nd bses cnnot be mtched you must use. Mtch the method to ech given question. Then solve.. 4 = 8 5 b. + = 74 c. e + 5 = 4. Prctice mtching bses method: 5. = b. 5 = 5 5 c = 50 d. 4 = 56 e. +.5 = 64 f. + = 4

4 4 U n i t 8 AdvF Dte: Nme: 5. Here is the generl eqution for eponentils tht will most often be used for eponentil word problems tht hve horizontl sympote of y = 0. Eplin the significnce of EACH letter in the contet of word problems nd summrize how to find the b in the eqution. ( ) p y = b 6. Solve the following problems by using the tril & error method. (you will lter lern how to use logs to solve these without the use of tril nd error). A drug s effectiveness decreses s time psses. Ech hour the 50mg drug loses 5% of its effectiveness. How long will it tke for the dose to rech the low level of 5mg? b. Crbon-4 hs hlf life of 570 yers. (If no initil mount is given, ssume 00% is the initil mount) Some pre-historic cve pintings were discovered in cve in Frnce. If the pint contined 48% of the orinil crbon-4, estimte the ge of the pinting. c. A cottge is originlly bought for $ If the vlue of this cottge pprecites t the rte of 7% per yer, when will the cottge be worth $00 000? d. The 00 fruit fly popultion doubles every 5 dys. In how mny dys is the popultion up to 000 flies? 4

5 5 U n i t 8 AdvF Dte: Nme: Wht is Logrithmic Function?. You ve seen inverses for ll the functions you ve studied so fr, ecept for eponentils. Recll tht inverse grphs re just reflections in y= lines nd inverse equtions hve =input nd y=output switched. recll inverses of lines nd qudrtics. Sketch the inverses of these eponentils b. Find the inverse equtions of these functions.. The inverse equtions you ve found bove don t hve the output isolted nd hence cnnot be written in function nottion. This is one of the resons tht logrithms were invented. Summrize the rule of switching eponentil form to logrithmic form or vise vers, then write down the inverse functions using function nottion for the bove questions.. Prctice switching the form. r = log p q b. b = c c. log 4 = d. = 5 5 5

6 6 U n i t 8 AdvF Dte: Nme: 4. Wht mening doeslog hve? 5. Evlute the following. b. log 6 c. d. log 64 e. log 9 f. 7 log Find the inverse functions y = log ( ). b. 4 y = y = log + 5 d. y = ( ) c

7 7 U n i t 8 AdvF Dte: Nme: Eponentil & Logrithmic Functions. Review how to find the eqution of the eponentil function from tble or grph. b. y Horizontl symptote t y=-4.. Summrize the steps of sketching eponentils. k( d) y = b + c Sketch the following functions, stte domin nd rnge +. y = (4) + 4. y = 00(0.5) 5. 8 y = 5(.5) 0 7

8 8 U n i t 8 AdvF Dte: Nme: 6. Summrize the steps of sketching logrithms: y = log [ k( d)] + c b Sketch the following functions, stte domin nd rnge 7. y = log 0. 6 y = log (6 + ) y = log (5 ) + 0. y = + log (4 + )

9 9 U n i t 8 AdvF Dte: Nme: Lws of Logrithms Red nd understnd the following proofs to the new logrithm lws.. Proof for log = let f ( ) = then f ( ) = log you lredy know tht f f = ( ( )), therefore log ( ) =. Proof is similr for log =. Proof for log ( y) = log + log y Use the eponent multipliction lw: = m n m+ n m let = log = m n nd y = log y = n log ( y) m n = log ( ) = log = m + n m+ n = log + log y Notice bses hve to mtch nd coefficients s well. clog ( y) = clog + clog y 4. Proof is similr for log = log log y y 5. Proof for n log ( ) = n log m ( ) { ( ) m ( ) Use the eponent power of power lw log n m let = log = m = log = log = mn = n log = n mn mn n 6. Proof for CHANGE of BASE log logb = log b { b = log = b log b = log log b = log log = log b log = b log = log b b 7. Proof for log = 0 Use the zero eponent lw 0 = 0 log = log 0 = log 9

10 0 U n i t 8 AdvF Dte: Nme: 8. Epnd the following by using lws of logs y. log z b. log y z c. log yz d. log y 4 9. Condense the following into single logrithm. 4 log log y b. 5log + log y + log z c. log z + log y d. log log y + log z 5 0. Evlute. log 9 (possible without clcultor since bses cn be mtched) b. 8 log 9 (not possible without the chnge of bse formul nd clcultor) 0

11 U n i t 8 AdvF Dte: Nme: Solving Eponentil Equtions by using Logs. An investment of $500 is invested in n ccount tht pys 6.4% compound nnully. How long will it tke for the originl mount of the investment to triple?. A culture of bcteri triples every 0 minutes. How long will it tke culture originlly consisting of 40 bcteri to grow to popultion of bcteri? Here re few strtegies to try when solving eponentil equtions Mtch bses (gol is to hve single bse on one side of the eqution nd the SAME single bse on the other side of the eqution no coefficients) If ppers once Switch forms to log If ppers more thn once, no plus/minus between bses use lws of eponents to rerrnge (remember if eponents re the sme you cn combine bses) If ppers more thn once, WITH plus/minus between bses Mke substitution to simplify the eqution. Solve the following equtions.. 5 = b = 4 4 +

12 U n i t 8 AdvF Dte: Nme: c = = 4 d. ( ) e. = = f = g h. =

13 U n i t 8 AdvF Dte: Nme: Solving Logrithmic Equtions When you re re solving logrithmic equtions, keep in mind the domin of logrithms nd discrd ny solutions tht mke log( zero) = undefined or log( negtive ) = undefined Also remember tht you CANNOT distribute the log over seprte terms, just like you CANNOT distribute eponents over severl terms E. Here re few strtegies to try when solving logrithmic equtions One isolted log Switch forms to ep Two logs of sme bse on either side Equte the inputs of the logs Mny logs Use lws of logs to condense Solve the following equtions. 6. log = 4 8 log ( ) = log ( 6 ) b..5.5 c. log 48 log 4.5 = log 49.6 d. log 5( + 6) + log 5( 6) =

14 4 U n i t 8 AdvF Dte: Nme: e. log 8 = f. 4 log 9 9 = log log g. log + log = h. log + log( ) = log0 4

15 5 U n i t 8 AdvF Dte: Nme: Solve Problems The logrithms re used in severl different rel life pplictions Erthquke mgnitudes (developed by Chrles F. Richter) Intensity of sound wves (how loud things re) ph scle M log I = I 0 = the mgnitude of the erthquke on the Richter scle = the intensity/mplitude of the reference erthquke = the intensity/mplitude of the wve detected by the seismogrph of the erthquke being mesured. In October of 005, Pkistn eperienced n erthqute of mgnitude 7.6 resulting in the deth of over people. Lter on tht month, Owen Sound eperienced n erthquke of 4. in mgnitude. How mny times more intense ws the Pkistn erthquke to the quke in Owen Sound? L 0log I = I 0 = the loudness of sound in decibels = the intensity of sound power per unit re (wtts/m ) of the threshold of hering = the intensity of sound power per unit re (wtts/m ) being mesured. How mny times more loud is rock concert with sound intensity of db thn the threshold of sound? 5

16 6 U n i t 8 AdvF Dte: Nme: ph = log( H + ) = the cidity of the substnce = the concetrtion of hydrogen ions (mol/l). A fish tnk s wter ws recently chnged with distilled wter of ph 7. The dy fter it ws chnged, pple juice ws spilled into it which cused the ph to drop to 5.8. By wht fctor hs [H + ] chnged? 4. A rdioctive substnce hs hlf-life of dys. Suppose you hve g of this substnce now. In how mny dys will the mss be. 0 -? 5. A bot sells t $ Ech yer it deprecites (decreses in vlue) by 5%. In how mny yers will the bot s vlue be $0 000? 6. Find the ph of solution with hydronium ion concentrtion of Find the decibel rting of sound with intensity of 5000I o 8. If sound hs decibel rting of 85, how much more intense is it thn the threshold sound? 6