MATH 181-Exponents and Radicals ( 8 )
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1 Mth 8 S. Numkr MATH 8-Epots d Rdicls ( 8 ) Itgrl Epots & Frctiol Epots Epotil Fuctios Epotil Fuctios d Grphs I. Epotil Fuctios Th fuctio f ( ), whr is rl umr, 0, d, is clld th potil fuctio, s. Rquirig th s to positiv would hlp to void th compl umrs tht would occur y tkig v roots of gtiv umrs. (E., ( ), which is ot rl umr.) Th rstrictio is md to clud th costt fuctio f ( ). Empl: f ( ), f ( ) ( ), 6 f ( ) (. 7) *Th vril i potil fuctio is i th pot. II. Grphig potil fuctio A. Plottig poits B. Grphig clcultor * Try f ( ) ( )
2 Mth 8 S. Numkr III. Applictio i Empl: Compoud Itrst Formul, A P( ) r for compoudig pr yr: or A P( ) for cotiuous compoudig: A P rt. totl of $,000 is ivstd t ul rt of 9%. Fid th lc ftr yrs if it is compoudd qurtrly: r t. ( ) A P( ), 000( 0 09 ) 8, 76.. totl of $,000 is ivstd t ul rt of 9%. Fid th lc ftr yrs if it is compoudd cotiuously: rt 0. 09( ) A P, 000 8, cotiuous compoudig yilds itrst: =9.6 IV. Th Numr A( ) ( ), s th gts lrgr d lrgr, th fuctio vlu gts closr to. Its dciml rprsttio dos ot trmit or rpt; it is irrtiol. I 7, Lord Eulr md this umr. You c us th ky o grphig clcultor to fid vlus of th potil fuctio f ( ). Empl: Fid 0,., 0, 00. 8, t t
3 Mth 8 S. Numkr Epots I. Evlut potil prssios. A. For y positiv itgr, tims such tht is th s d is th pot Empl: = 7 B. For y ozro rl umr d y itgr, 0 Ad C. I multiplictio prolm, th umrs or prssios tht r multiplid r clld fctors. If c, th d r fctors of c. Empls: ( ) 0 c. ( ) d. =. = D. Proprtis of Epots m m m ( m) such tht 0 ( ) ( ) m m m m m ( ) m m such tht 0 m
4 Mth 8 S. Numkr Empls:. ( ). y y c. y y 6 d. ( ). m m c f. ( ) 6 c = II. Frctiol Epots Dfiitio I.: such tht is clld th id d is th rdicd. m Dfiitio II: ( ) m m * m is itgr, is positiv itgr, d is rl umr. If is v, 0. Empls: 9 6 ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( )( )
5 Mth 8 S. Numkr Simplst Rdicl Form Evlut squr d highr roots A. Rdicl ottio: A umr c is sid to th root of if c. Squr root: if Cu root: if If giv, th symol is clld rdicl, is th id, d is th rdicd. Th positiv root is clld th pricipl root. To dot gtiv root, us, 8, tc. Empls: 6 6, 6 6, ( ) 8, 6 is ot rl umr * Wh is gtiv d is v, is ot rl umr. B. Simplifyig Rdicl Eprssios If d r rl umrs or prssios for which th giv roots ist. m, r turl umrs. Hr r som proprtis of rdicls.. If is v,. ( 0 ). If is odd,. m. ( ) m
6 Mth 8 S. Numkr Empls: Simplify.. 8c d = c. 6 m 6 d. 0m m. 7 9 f. 9 8 = g. 6 h. = 7 i. 8 j. 8 k l. 6 Empls: Simplify.. 8c d = c. 6 m 6 d. 0m m 6
7 Mth 8 S. Numkr Rtioliz th umrtor: y y y ( ) ( y) y y y y C. Rtiol Epots For y rl umr d y turl umrs m d, m/ / m m/ m/ m Empls:Covrt pot prssios to rdicl ottios / / = c / = Empls: Covrt to potil ottio d simplify.. ( 7y) ( 7y) /. 6 c. 7 d. ( ). = f. 7= D. Rltioship tw v d odd roots r s follows: Lt rl umr d positiv itgr grtr th o. I prticulr,. d I grl,. If is v positiv itgr, d. If is odd positiv itgr, 7
8 Mth 8 S. Numkr Empls: Writ ch prssio i simplst rdicl form. Assum ll vrils rprst positiv rl umrs y 0y 6 7 y 6 z 7 y 0 z 6 y 8 y 8
9 Mth 8 S. Numkr Logrithms Logrithmic Fuctios d Grphs I. Logrithm Logrithmic, or logrithm, fuctios r ivrs of potil fuctios d hv my pplictios. Empl: Covrt ch to logrithmic qutio: 6 >log log t 70 log 70 t Empl: Fid ch of th followig logrithms: log, log log 8 log 9 log 6 log 8 8 * log 0 d log, for y logrithmic s 9
10 Mth 8 S. Numkr II. Commo Logrithm d Nturl Logrithm A. Commo logrithms r of s-0. Th rvitio log, with o s writt, is usd to rprst th commo logrithms, or s 0 logrithms. Empl: log 9 ms log 0 9 B. Nturl Logrithms log 00 = log 0 00 Logrithms, s, r clld turl logrithms. Th turl logrithm's rvitio is l. Empl: l ms log III. Chgig Logrithmic Bss log M log log 0 0 M l M l Empl: log log log l 6. 8 log 6 l
11 Mth 8 S. Numkr IV. Proprtis of Logrithms A. y y Empl: 8 B. y y Empl: C. ( ) y y Empl: ( ) 096 D. log y log log y Empl: log log ( 7 9) log 7 log 9 E. log ( ) log log y y 6 Empl: log ( ) log 6 log 8 8 F. log ( ) log G. log 0 Empl: log ( ) log 6 Empl: log 8 0
12 Mth 8 S. Numkr H. log Empl: log 8 8 I. log ( ) Empl: log ( ) J. f ( ) log l, 0 is th turl logrithmic fuctio K. l L. l l M. l 0 N. l( uv) l u l v V. Try ths: A. Eprss ch s sum, diffrc, or multipl of Logrithms. log. log. log 9. log 6. log 6. log log ( ) 8. log 7 y 9. log ( ) 8 0. log 0 y. l 7. log 7
13 Mth 8 S. Numkr B. Eprss ch s th logrithm of sigl qutity. log log c. log 9 log. log log. log log. l ( ) 6. l l y l z
14 Mth 8 S. Numkr Solvig Epotil d Logrithmic Equtios I. To solv potil qutio, first isolt th potil prssio, th tk th logrithm of oth sids d solv for th vril. Empl: Solvig Solvig 7 l l 7 l Solvig 60 l l l. 0 Solvig 0 ( )( ) 0, if l l l. 69 if l l 0 Try solvig: II. To solv logrithmic qutio, rwrit th qutio i potil form (potitig) d solv for th vril. l Empl: Solvig l Solvig l l l l. 607 l Solvig l l. 6 Solvig l l( ) l ( ) *l * l
15 Mth 8 S. Numkr Try solvig: log( ) log( ) l( 00) l log log( ) log log( ) l l8 l l( ) log( ) log
16 Mth 8 S. Numkr III. Applictio Empl: You hv dpositd $00 i ccout tht pys 6.7% itrst, compoudd cotiuously.. How log will it tk your moy to doul?. How log will it tk your moy to tripl? rt. for cotiuous compoudig, A P To fid th tim rquird to doul, A t. 067t l l l. 067t l t To fid th rquird to tripl, A t. 067t l l l. 067t l t. 067 IV. Epotil d Logrithmic Modls I. Th fiv most commo typs of mthmticl modls ivolvig potil fuctios d logrithmic fuctios:. Epotil growth: y, 0. Epotil dcy: y, 0. Gussi modl: y ( ) c t t t This typ of modl is usd i proility d sttistics to rprst popu ltios tht r ormlly distriutd. Th stdrd orml distriutio tks th form y Th grph of Gussi modl is ll-shpd curv.. Logistics growth modl: y ( y = pop. siz, = tim) ( c) d 6
17 Mth 8 S. Numkr Som popultio iitilly hv rpid growth, followd y dcliig rt of growth. This typ of growth pttr is of logistics curv; it is lso clld sigmoidl curv. Empl: ctri cultur.. Logrithmic modl: y l( ), y log ( ) E.: O th Richtr scl, th mgitud of R of rthquk of itsity 0 I: R I log 0 I 0 II. Tk look t th sic shps of ths grphs:. y. y. y. y. y l 6. y log 0 7
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