1 . EXPONENTIAL FUNCTIONS & THEIR GRAPHS EXPONENTIAL FUNCTIONS EXPONENTIAL nd LOGARITHMIC FUNCTIONS re non-lgebric. These functions re clled TRANSCENDENTAL FUNCTIONS. DEFINITION OF EXPONENTIAL FUNCTION The EXPONENTIAL FUNCTIONf with bse is denoted f ( )= where > 0, nd is rel number. WHY DO WE WANT? WHY DO WE WANT > 0? EXAMPLE # Evlute.. A. B C. π D. 5 7 E. ( 0.6) GRAPHS OF EXPONENTIAL FUNCTIONS f ( ) = g( ) = - h( ) = k( ) = All of the rules for reflecting, stretching & compressing, nd trnslting still hold for ll EXPONENTIAL FUNCTIONS.
2 Grph of y =, > Grph of y =, > 0 s DOMAIN RANGE INTERCEPT INCREASING/DECREASING ASYMPTOTE 0 s + Continuous CONTINUOUS?... Continuous GRAPH y = y = - ( ) EXAMPLE # Compre the grphs of g to the grph of f =. A. g( ) = B. g( ) = + C. g( ) = D. g( ) = THE NATURAL BASE e The irrtionl numbere is clled the NATURAL NUMBER. f = e is the NATURAL EXPONENTIAL FUNCTION. The function ( ) The numbere cn be pproimted by + for relly LARGE vlues of s ± EXAMPLE # Evlute. A. e B. e EXAMPLE # Sketch the grph of ech eponentil function. 0. A. f ( ) = e B. g( ) = e
3 COMPOUND INTEREST Suppose you invest some money (principl,p ) t n nnul interest rte (r ), compounded once yer. This pttern is repeted ech yer s follows: TIME IN YEARS END OF YEAR BALANCE 0 P = P P = P + r ( ) ( ) ( )( ) ( ) P = P + r = P + r + r = P + r P = P + r n ( ) n n But interest is not lwys compounded yerly. So to ccommodte more frequent (qurterly, monthly, weekly, or dily) compounding of interest, letn be the number of compoundings per yer nd lett be the number of yers. Then the interest rte per compounding period is r n. The formuls below follow this pttern. FORMULAS FOR COMPOUND INTEREST Aftert YEARS, the BALANCEA in n ccount with PRINCIPALP nd ANNUAL INTEREST RATEr (epressed s deciml) is given by the following formuls: r. Forn COMPOUNDINGS per YEAR: A = P + n. For CONTINUOUS COMPOUNDING : A = Pe rt nt EXAMPLE #5 A totl of $,000 is invested t n nnul interest rte of 9%. Find the blnce in the ccount fter 5 yers if it is compounded A. qurterly. B. monthly. C. continuously. EXAMPLE #6 Let y represent the mss of quntity of rdioctive element whose hlf-life is 5 yers. t 5 Aftert yers, the mss (in grms) is y = 0. A. Wht is the initil mss? B. How much of the initil mss is present fter 80 yers?
4 . LOGARITHMIC FUNCTIONS & THEIR GRAPHS LOGARITHMIC FUNCTIONS Recll tht for function to hve n inverse, it must be ONE-TO-ONE nd therefore pss the HORIZONTAL LINE TEST. If you tke look bck t the grphs of EXPONENTIAL FUNCTIONS, you will notice tht every function of the form, > 0, psses the Horizontl Line Test nd therefore must hve n inverse. These inverse functions re clled LOGARITHMIC FUNCTIONS. DEFINITION OF LOGARITHMIC FUNCTION For > 0 nd 0<, y = log if nd only if = The function f ( ) = log is clled the LOGARITHMIC FUNCTION WITH BASE. y RECALL THAT TO FIND AN INVERSE, YOU MUST SWITCH X AND Y Emple: Find the inverse of = y. REMEMBER THAT A LOGARITHM IS AN EXPONENT COMMON LOGARITHM Any logrithm with BASE 0. f ( ) = log0 EXAMPLE # Evlute. A. log B. log 8 C. log 00 D. log.5 E. log.. EXAMPLE # Evlute. A. log 0 B. 0 log.5 C. log( )
5 The following properties follow DIRECTLY from the DEFINITION of logrithmic functions... PROPERTIES OF LOGARITHMS = becuse = 0 log 0 log = = log becuse = = inverse property log nd. log = log y, then = y one-to-one EXAMPLE # Solve for. A. log 5 = B. log 5= log ( + ) C. log = D. log7 = GRAPHS OF LOGARITHMIC FUNCTIONS Since the logrithms re inverses of eponentil functions, we cn use wht we know bout inverses nd eponentil functions to describe some properties of the grphs of logrithms Grph of y =, > Grph ofy = log, > (, + ) DOMAIN ( 0, + ) RANGE ( 0, ) INTERCEPT Incresing -is( y = 0) 0 s INCREASING/DECREASING ASYMPTOTE log s 0 Continuous CONTINUOUS?... Continuous + GRAPH y = 5 y = - y = log Reflection of grph of y = in the line y =
6 EXAMPLE # Compre the grphs of g to the grph of f ( ) = log. A. g( ) = log( + ) B. g( ) = log C. g( ) = + log D. g( ) = log( ) THE NATURAL LOGARITHMIC FUNCTION The logrithmic function with bsee is the NATURAL LOGARITHMIC FUNCTION. THE NATURAL LOGARITHMIC FUNCTION The function defined by f = = > ( ) log ln, 0 e is clled the NATURAL LOGARITHMIC FUNCTION. EXAMPLE #5 Evlute. A. ln B. ln( ) C. ln D.ln 0 5 PROPERTIES OF NATURAL LOGARITHMS... becuse e 0 ln = 0 = e = becuse e = e ln ln ln e = becuse e =. ln = ln y, then = y EXAMPLE #6 Rewrite ech epression using the properties of nturl logrithms. A. ln e B. ln e C. 0 ln e D. ln e EXAMPLE #7 Find the domin of ech function. A. f ( ) = ln( ) B. g( ) = log( + ) C. g( ) = log
7 . PROPERTIES OF LOGARITHMS CHANGE OF BASE CHANGE-OF-BASE FORMULA Let, b, nd be positive rel numbers, such tht nd b. Then, log cn be converted to different bse using ny of the following: Bse b Bse 0 Bse e log logb log ln = log = log = log log ln b PROPERTIES OF LOGARITHMS PROPERTIES OF LOGARITHMS Let be positive rel number, such tht, nd letn be rel number. Ifu nd v re positive rel numbers, the following re true: log uv = log u + log v. ( ). log u = log u log v v n. log ( ) u = n log u EXAMPLE # Evlute. A. log 7 use bse 0 B. log use bsee EXAMPLE # Epnd the epression. y A. log B. ln y C. log ( + )
8 EXAMPLE # Condense the epression to single logrithm. A. log 8+ log z B. log log( + ) C. ln + ln( + 5) ln( 5) D. ln( + ) ln + ln( + 5) + ln( + 5)
9 . SOLVING EXPONENTIAL & LOGARITHMIC EQUATIONS STRATEGIES FOR SOLVING. Rewrite the eqution in form to use the ONE-TO-ONE properties of eponentil & logrithmic functions. y = if nd only if = y log = log y if nd only if = y. Rewrite n EXPONENTIAL eqution in LOGARITHMIC FORM & pply the INVERSE PROPERTY of logrithms. log. Rewrite LOGARITHMIC eqution in EXPONENTIAL FORM & pply the INVERSE PROPERTY of eponentil functions log = = EXAMPLE # Solve for. A. = 7 B. ln ln = 0 C. e = 5 D. ln = E. log5 = F. log = SOLVING EXPONENTIAL EQUATIONS EXAMPLE # Solve for. A. 7 = B. 7 e = C. ( ) = D. e = 5
10 EXAMPLE # Solve for. = B. 7 e = 5 5 A. ( ) EXAMPLE # Solve for : e e + = 0 EXAMPLE #5 Solve fort : t =
11 SOLVING LOGARITHMIC EQUATIONS BE SURE TO CHECK YOUR SOLUTIONS SINCE LOGARITHMS ARE ONLY DEFINED FOR VALUES > 0 POSITIVE NUMBERS EXAMPLE #6 Solve for. = B. log ( 5 ) = log ( + 7) C. ( ) A. ln ln + = EXAMPLE #7 Solve for : 5+ ln = EXAMPLE #8 Solve for. A. ln( ) + ln( ) = ln B. ( ) ( ) ln + ln = ln EXAMPLE #9 Solve for : ( ) ( ) log + + log + =
12 EXAMPLE #0 You hve deposited $500 in n ccount tht pys 6.75% interest, compounded continuously. How long will it tke you to double your money? EXAMPLE # A deposit of $5000 is plced in svings ccount for yers. The interest for the ccount is compounded continuously. At the end of two yers, the blnce in the ccount is $5,6.50. Wht is the nnul interest rte for this ccount? EXAMPLE # You hve deposited $500 in n ccount tht pys 0% interest, compounded qurterly. How long will it tke you to triple your money?
13 .5 EXPONENTIAL AND LOGARITHMIC MODELS COMMON EXPONENTIAL AND LOGARITHMIC MODELS. EXPONETIAL GROWTH MODEL: f( ) = e. GAUSSIAN MODEL: f( ) = e b y = e, b > 0 y = ( ) b c e. EXPONENTIAL DECAY MODEL: f( ) = e -. LOGISTIC GROWTH MODEL: - b y = e, b > 0 - f( ) = y = + be r +e LOGARITHMIC MODELS: f( ) = +ln( ) f( ) = +log( ) y = + b ln
14 EXAMPLE # A rdioctive substnce hs hlf-life of 0 yers. How much remins of oz. smple fter 00 yers? EXAMPLE # How mny yers will it tke your money to double if you deposit it into fund pying % compounded dily? EXAMPLE # How much money must be invested t 5% interest, compounded continuously, to yield $650 fter yers? EXAMPLE # An initil deposit of $800 is mde into svings ccount for which the interest is compounded continuously. The blnce will triple in 8 yers. Wht is the nnul interest rte for this ccount?
15 EXAMPLE #5 If the verge time between successive occurrences of some event isλ, then the probbility of witing less thnt units of time between successive occurrences cn sometimes be pproimted by the model t λ F ( t) = e The verge time between incoming clls t switchbord is minutes. If cll hs just come in, find the probbility tht the net cll will be within A. minute B. 5 minutes PROPERTIES OF EXPONENTS EXAMPLE #6 Solve for : 5 =