Logarithms. Secondary Mathematics 3 Page 164 Jordan School District
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1 Logarithms Sondary Mathmatis Pag 6 Jordan Shool Distrit
2 Unit Clustr 6 (F.LE. and F.BF.): Logarithms Clustr 6: Logarithms.6 For ponntial modls, prss as a arithm th solution to a and d ar numrs and th as is, 0 or..6 Evaluat th arithm using thnoy..6 Undrstand th invrs rlationship twn ponnts and arithms..6 Us th rlationship twn ponntials and arithms to solv prolms involving arithms and ponnts. t d whr a,, f, 0, f, 0, 0,,,0, Domain:, Rang: 0, Horizontal Asymptot: y 0 Intrpt: 0, End Bhavior: lim f ; lim f 0 Domain: 0, Rang:, Vrtial Asymptot: 0 Intrpt:,0 End Bhavior: lim f lim f 0 ; Eponntial and arithmi funtions ar invrss of ah othr. Two of th most widly usd arithms ar th ommon, whih is as 0, and is writtn as 0 and th natural, whih is as, and is writtn as ln. Dfinition of a Logarithm ln if and only if if and only if Sondary Mathmatis Pag 6 Jordan Shool Distrit
3 Eampl : Rwrit ah of th following in ponntial form Th as is and th ponnt is. Th as is and th ponnt is Th as is 6 and ponnt is 0. Eampl : Rwrit ah of th following in arithmi form Th as is and th ponnt is. Th as is 0 and th ponnt is Th as is 6 and ponnt is. Basi Proprtis of Logarithms whr 0,, 0, and is any ral numr. 0. ln 0.. ln.. ln.. ln Sondary Mathmatis Pag 66 Jordan Shool Distrit
4 Eampl : Us th proprtis of arithms to valuat th prssion without a alulator. 0 ln6. d Us of asi proprty. ln6 6 0 Us of asi proprty. Us of asi proprty. d. 0 0 Us of asi proprty. Prati Eriss A Rwrit ah of th quations in ponntial form Rwrit ah of th quations in arithmi form Us th proprtis of arithms to valuat th prssion without a alulator ln ln. ln 0 Th Prinipl of Eponntial Equality For any ral numr, whr, 0, or, is quivalnt to. In othr words, powrs of th sam as ar qual if and only if th ponnts ar qual. Sondary Mathmatis Pag 67 Jordan Shool Distrit
5 Eampl : Solv th following Rwrit in ponntial form and solv for. Rwrit in ponntial form and solv for. Rwrit in ponntial form and solv for. Eampl : Solv th following Chang th ass on oth sids of th quation so that thy ar th sam as. Rwrit using ponnt ruls. Simplify using ponnt ruls. Solv using th prinipl of ponntial quality. Rwrit in ponntial form and solv for. Sondary Mathmatis Pag 68 Jordan Shool Distrit
6 Prati Eriss B Solv th following quations , For ah of th following ruls, Produt Rul,, y, and ar ral numrs. y y Quotint Rul ln y ln ln y y Powr Rul y ln ln ln y y ln ln Eampl 6: Epand th following prssions. a ln mn. wh a a a a Us th produt and quotint ruls to rwrit th prssion. Us th powr rul to rwrit th prssion. Sondary Mathmatis Pag 69 Jordan Shool Distrit
7 . ln ln mn mn ln m ln n ln m ln n ln m ln n wh a w h a w h a w h a Us th powr rul to rwrit th prssion. Us th produt rul to rwrit th prssion. Us th powr rul to rwrit th prssion. Us th distriutiv proprty. Us th produt and th quotint ruls to rwrit th prssion. Us th powr rul to rwrit th prssion. Us th distriutiv proprty. Not: Whn using th quotint rul, all trms in th dnominator will sutratd. Eampl 7: Condns th following prssions. ln ln a. ln a ln 7 ln ln d ln ln ln ln ln Us th powr rul to rwrit th prssion. Us th quotint rul to rwrit th prssion. a / a a Us th powr rul to rwrit th prssion. Us th produt and quotint ruls to rwrit th prssion Sondary Mathmatis Pag 70 Jordan Shool Distrit
8 . ln a ln 7ln ln d 7 ln a ln ln ln d ln a 7 d Us th powr rul to rwrit th prssion. Us th produt and quotint ruls to rwrit th prssion. Prati Eriss C Epand th following prssions.. y y. 7 9yz a 6. y 9. a ln y z Condns th following prssions. 0. ln ln ln 7ln y ln z. y z. z y ( ) ( ) ln ln ln Sondary Mathmatis Pag 7 Jordan Shool Distrit
9 Chang of Bas Formula for Logarithms ln Most alulators only hav and. In ordr to valuat arithms with a diffrnt as, you will nd th hang of as formul, or ln, ln Eampl 8: Find an approimation for th following prssions. ln d. 6 ln 7.76 Us th alulator Us th alulator ln7 7.0 ln 6 d ln ln Us th hang of as formula and your alulator. Us th hang of as formula and your alulator. Prati Eriss D Find an approimation for th following prssions.. ln ln Sondary Mathmatis Pag 7 Jordan Shool Distrit
10 Eampl 9: Find th domain of th funtion thn graph it. f f 0, Th domain is. Th domain of a arithmi funtion has to gratr than zro. Us th hang of as formula to ntr th funtion. y or ln y ln 0 Th domain is,. Th domain of a arithmi funtion has to gratr than zro. Us th hang of as formula to ntr th funtion. y or ln y ln Sondary Mathmatis Pag 7 Jordan Shool Distrit
11 Prati Eriss E Find th domain of th funtion and thn graph it.. f. f 6. f. f 7. f 6. f 7. f 8. f 7 9. f f. f 6. f Th Prinipl of Logarithmi Equality 0 For any arithmi as,, and for any and is quivalnt to y. In othr words, two prssions ar qual if and only if th arithms of thos prssions ar qual. y 0, y Eampl 0: Solv ah quation.. ln ln ln d ?? Writ th quation in ponntial form. Solv for. Chk your answr in th original quation. Sondary Mathmatis Pag 7 Jordan Shool Distrit
12 ln ln 0.0 ln ln Isolat th ponntial trm. Find th natural arithm of oth sids. Us th proprty as. Solv for. ln to liminat th. ln ln ln ln ln 0 ln ln ln ln ln ln 8 ln ln? is an tranous solution.? Us th produt rul and th powr rul to rwrit oth sids of th quation. Epand oth sids of th quation. Us th prinipal of arithmi quality to liminat th arithm. Solv for. Chk oth answrs in th original quation. Th ln 8 and ln ar undfind. ln ln ln ln ln 8 ln 0 ln 8 ln? ln ln is a solution.??? ln ln Sondary Mathmatis Pag 7 Jordan Shool Distrit
13 d. ln ln ln ln ln ln ln ln ln ln ln ln 0.7 ln Find th natural of oth sids. Us th powr rul to rwrit th prssion. Us th distriutiv proprty. Solv for. Prati Eriss F Solv ah quation Eampl : Find th invrs of ah funtion. f ln f 6. f d. f Sondary Mathmatis Pag 76 Jordan Shool Distrit
14 ln y f y ln ln y ln y y y Sustitut ah with y and y with. Isolat th arithmi trm. Us th proprty arithm. Solv for y. ln to liminat th y y y y y y f y Sustitut ah with y and y with. Isolat th arithmi trm. Us th proprty arithm. Solv for y. to liminat th. 6 y6 y6 y6 y y f y Sustitut ah with y and y with. Isolat th ponntial trm. Us th proprty as of th ponnt. Solv for y. to liminat th Sondary Mathmatis Pag 77 Jordan Shool Distrit
15 d. y y y y y y y f y Sustitut ah with y and y with. Isolat th ponntial trm. Us th proprty as of th ponnt. Solv for y. to liminat th Eriss G Find th invrs of ah funtion.. f 7. f 0. f 6 ln 8. f. f 7 6. f 7. f 8. f 9. f f. f. f 7 Sondary Mathmatis Pag 78 Jordan Shool Distrit
16 Using th Strutur of Eprssions to Solv Equations (Honors) Eampl : Solv th quation 8 0. u 8 0 u8 0 u u 6 0 u 60 u 6 6 ln ln 6 ln 6 ln 6 u 0 u Th quation is quadrati in natur, lt. Rwrit th quation in trms of u. u Solv for u. Sustitut u and solv for. An ponntial funtion will nvr qual a ngativ numr. Prati Eriss H Solv ah quation Sondary Mathmatis Pag 79 Jordan Shool Distrit
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