Chapter 3 Exponential and Logarithmic Functions Section 3.1


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1 Chpter 3 Eponentil nd Logrithmic Functions Section 3. EXPONENTIAL FUNCTIONS AND THEIR GRAPHS Eponentil Functions Eponentil functions re nonlgebric functions. The re clled trnscendentl functions. The eponentil function f with bse is denoted b f ( ) = where > 0,, nd is n rel number. Grphs of Eponentil Functions Complete the tble of vlues for ech eponentil function nd sketch the grph of ech function. f ( ) = f ( ) = f() f() h( ) = m( ) = f() f() Bsic Chrcteristics of Eponentil Functions Grph of =, Domin: > Rnge: YIntercept: Incresing is( = 0) is horizontl smptote ( 0 s ) continuous
2 Chpter 3 Eponentil nd Logrithmic Functions Section 3. Grph of =, > Domin: Rnge: YIntercept: Decresing is ( = 0 )is horizontl smptote ( 0 s ) continuous Eplortion: Use grphing utilit to grph = with = 3, 5, nd 7 in the sme viewing window. (Use viewing window in which  < < nd 0 < <.) How do the grphs compre with ech other? Which grph is on the top in the intervl (,0)? Which is on the bottom? Which grph is on the top in the intervl (0, )? Which is on the bottom? Repet this eperiment with the grphs of 0 < <.) Wht cn ou conclude bout the shpe of the grph of = for = 3, 5, nd 7. ( Use viewing window in which  < < nd = nd the vlue of? Trnsformtions of Eponentil Functions All the trnsformtion ou lerned in Chpter One cn be pplied here. f ( ) Emple #: Use the eponentil function bove nd its tble of vlues to sketch the grphs of the following eponentil functions. + f ( ) = g( ) = h( ) = k( ) =
3 Chpter 3 Eponentil nd Logrithmic Functions Section 3. j( ) = l( ) = + m( ) = + 3 Using the grphs of the eponentil functions in emple #, complete the following tble. g( ) = + h( ) = k( ) = j( ) = l( ) = + m( ) = 3 Asmptotes Domin Rnge Intercepts Incresing or Decresing Nturl Eponentil Function The irrtionl number e is clled the nturl bse. e The function f ( ) = e is the nturl eponentil function. Complete the tble of vlues to mke sketch for the grph of f ( ) = e f ( ) = e Grph of f ( ) = e Domin: Rnge: intercept: smptote: Approimtion of the Number e Use grphing utilit to grph the following two functions: = + nd = e. 3
4 Chpter 3 Eponentil nd Logrithmic Functions Section 3. Wht do ou notice bout the grph of = + s? Compound Interest A principl P is invested t n nnul interest rte r, compounded once er nd the interest is dded to the principl t the end of the er. Wht will the blnce be t the end of ever er? Time in Yers Blnce After Ech Compounding 0 P=P P = P = 3 P 3 = t P t = Wht will the blnce be t the end of ech er if the number of compounded is more thn one? Let n = the number of compounding per er. Derivtion of Formul for Continuous Compounding Wht would hppen if ou let the number of compoundings n increse without bound? Formuls for Compound Interest After t ers, the blnce A in n ccount with principl P nd nnul interest rte r (epressed s deciml) is given b the following formuls. r. For n compoundings per er: A = P + n. For continuous compounding: A = Pe rt nt Emple#: Determine the blnce A t the end of 0 ers if $500 is invested t 6.5% interest nd the interest is compounded ) Qurterl b) Continuousl 4
5 Chpter 3 Eponentil nd Logrithmic Functions Section 3. Emple #3: Determine the mount of mone tht should be invested t 9% interest, compounded monthl, to produce finl blnce of $30,000 in 5 ers. Emple#4: A totl of $0,000 is invested t n nnul interest rte r. Find r grphicll if the blnce fter 5 ers of compounding continuousl is $8,800. Eponentil Growth Emple #5: The popultion of town increses ccording to the model with t = 0 corresponding to t P( t) = 500e where t is the time in ers, ) Find the popultion in 99, 995 nd 998. b) Use grphing utilit to grph the function for the ers 990 through 05 nd pproimte the popultion in 005 nd 05. c) Verif our nswers in prt c) lgebricll. Eponentil Dec Emple #6: Let Q (in grms) represent the mss of quntit of rdium 6, which hs hlflife of 60 ers. The t 60 quntit of rdium present fter t ers is Q( t) = 5. ) Determine the intitil quntit ( when t = 0). b) Determine the quntit present fter 000 ers. c) Use grphing utilit to grph the functions over the intervl t = 0 to t = d) When will the quntit of rdium be 0 grms? Eplin. 5
6 Chpter 3 Eponentil nd Logrithmic Functions Section LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Logrithmic Functions Does the eponentil function f ( ) =,0 < hve n inverse? Wh? Definition of Logrithmic Function (inverse of the eponentil function f ( ) =,0 < ) For > 0 nd 0<, = if nd onl if =. The function f ( ) = is clled the rithmic function with bse. Eponentil form Logrithmic form 00 = = 3 3 = = 6 Emple #: Evlute ech epression The rithmic function with bse 0 is clled the common rithmic function. Properties of Logrithms. 0 = becuse.. = becuse. 3. = nd =. (Inverse Properties) 4. If =, then =. (Onetoone propert)
7 Chpter 3 Eponentil nd Logrithmic Functions Section 3. Emple #: Solve ech eqution for. 9 = ln = ln 9 = Grphs of Logrithmic Functions Since = is the inverse function of in the line =. Sketch the grph of ech function. f ( ) = 3 nd g( ) = 3 =, we cn sketch the grph of = b reflecting the grph of = = Grph of =, > Domin: Rnge: Intercept: Asmptote: Incresing nd continuous Reflection of grph of = in the line = Trnsformtion of Grphs of Logrithmic Functions Once gin, ll the trnsformtions ou lerned in chpter one cn be pplied here. f ( ) = 0 Emple #3: Use the rithmic function bove to sketch the grphs of the following rithmic functions. g( ) ( ) = 0 h( ) 0 = + k( ) = 0
8 Chpter 3 Eponentil nd Logrithmic Functions Section 3. Use the grphs of the rithmic functions in emple #3 to complete the following chrt. g( ) = ( ) 0 Asmptotes Domin Rnge Intercepts Incresing or Decresing h( ) = + 0 k( ) = 0 The Nturl Logrithmic Function Everthing ou lerned bout the rithmic function will still ppl here. The onl difference is tht ou re working with bse e. The Nturl Logrithmic Function The function defined b f ( ) = e = ln, > 0 is clled the nturl rithmic function. Grph of = e = ln Domin: Rnge: Intercept: Asmptote: Incresing nd continuous Reflection of grph of = e in the line = The properties of nturl rithms re identicl to those of the common rithms. Properties of Nturl Logrithms 0. ln = 0 becuse e =. ln e = becuse e = e 3. ln e ln = nd e =. (Inverse Properties) 4. If ln = ln, then =. ( Onetoone propert) Sketch the grph of the nturl rithmic function. Determine the domin nd identif n verticl smptote nd  intercept h( ) = ln( ) Domin: Verticl Asmptote: intercept: 3
9 Chpter 3 Eponentil nd Logrithmic Functions Section 3. h( ) = ln(3 ) Domin: Verticl Asmptote: intercept: 4
10 Chpter 3 Eponentil nd Logrithmic Functions Section PROPERTIES OF LOGARITHMS Properties of Logrithms Let be positive number such tht, nd let n be rel number. If u nd v re positive rel numbers, the following properties re true.. ( uv) = u + v. ln( uv) = ln u + ln v u u. = u v. ln ln u ln v v v = n 3. u = n u 3. ln u = nln u Proofs for the properties of rithms: n Emple #: Epress the rithm in terms of ln nd ln 3. ln8 ln 8 Rewriting Logrithmic Epressions Eplortion: Use grphing utilit to grph the functions = ln ln( 3) nd = ln in the sme viewing 3 window. Does the grphing utilit show the functions with the sme domin? If so, should it? Eplin our resoning. Emple #: Epnd the rithm ln ln 4
11 Chpter 3 Eponentil nd Logrithmic Functions Section 3.3 Emple #3: Condense the rithm 3ln + ln 4ln z 4[ln z + ln( z + 5)] ln( z 5) ln 5t ln t 4 Emple #4: Find the ect vlue of the rithm without using clcultor. (If this is not possible, stte the reson.) ( 6) 5 3 ln e Emple #6: True or flse. ln ln ln = (ln )(ln ) = ln( + ) ln On our clcultor, it s possible to find rithms of bses 0 nd e. Suppose ou wnt to find the vlue of rithm of bse 7, wht cn ou do? Let 7 30 = z ChngeofBse Formul Let,b, nd be positive rel numbers such tht nd b. Then cn be converted to different bse using n of the following formuls. Bse b Bse 0 Bse e = b b = 0 0 ln = ln Emple #5: Evlute the rithm using the chngeofbse formul
12 Chpter 3 Eponentil nd Logrithmic Functions Section SOLVING EXPONENTIAL AND LOGARITHMIC EQUATIONS Strtegies for Solving Eponentil nd Logrithmic Equtions. Rewrite the given eqution in form to use the OnetoOne Properties of eponentil or rithmic functions OnetoOne Properties = if nd onl if = = if nd onl if =. Rewrite n eponentil eqution in rithmic form nd ppl the Inverse Propert of rithmic functions. Inverse Properties of Logrithmic Functions: = 3. Rewrite rithmic eqution in eponentil form nd ppl the Inverse Propert of eponentil functions. Solving Eponentil Equtions Inverse Properties of Eponentil Functions: Emple #: Solve ech eqution for. = 3 = = 4 64 ln ln = 0 0 = 0 = 4 3 e = e 5e + 6 = 0 Emple #: Simplif the epression. ln e ln e e 3 ln
13 Chpter 3 Eponentil nd Logrithmic Functions Section 3.4 Solving Logrithmic Equtions Emple #3: Solve for. ln = 7 0 = 6 ln 8 = 5 ln( + ) = ( 8) = ( + ) =
14 Chpter 3 Eponentil nd Logrithmic Functions Section EXPONENTIAL AND LOGARITHMIC MODELS Most Common Tpes of Mthemticl Models Eponentil growth model b = e, b > 0 Verticl smptote(s): = e Horizontl smptote(s): Eponentil dec model b = e, b > 0 Verticl smptote(s): = e Horizontl smptote(s): Gussin model ( b) = e c Verticl smptote(s): = 4e Horizontl smptote(s): Logistic growth model = Verticl smptote(s): r + be 3 = + e 5 Horizontl smptote(s): Logrithmic models = + b ln Verticl smptote(s): = + ln Horizontl smptote(s): = + b 0 = + 0 Verticl smptote(s): Horizontl smptote(s):
15 Chpter 3 Eponentil nd Logrithmic Functions Section 3.5 Modeling Popultion Growth The tble shows the popultion (in millions) of countr in 997 nd the projected popultion (in millions) for the er 00. (Source: U.S. Bureu of the Census) ) Find the eponentil growth model = e for the popultion in ech countr b letting t = 0 correspond to 997. Use the model to estimte the popultion of ech countr in 030. bt Countr Croti Mli Singpore Sweden b) You cn see tht the popultion of Mli nd Sweden re growing t different rtes. Wht constnt in the eqution bt = e is determined b these different growth rtes? Discuss the reltionship between the different growth rtes nd the mgnitude of the constnt. c) You cn see tht the popultion of Singpore is incresing while the popultion of Croti is decresing. Wht bt constnt in the eqution = e reflects this difference? Eplin.
16 Chpter 3 Eponentil nd Logrithmic Functions Section 3.5 Fitting Models to Dt The numbers (in millions) of vinl single records sold in the United Sttes in the ers 984 through 997 re listed below where t is the er, with t = 4 representing 984. Crete sctter plot of the dt. Decide which tpe of model best fits this dt. Then find the model. Eplin wh ou think the model ou chose is good fit to the dt. er (t) numbers (in millions) of vinl single records ()
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