5 Eponentil nd logrithmic functions 5A Inde lws 5B Negtive nd rtionl powers 5C Indicil equtions 5D Grphs of eponentil functions 5E Logrithms 5F Solving logrithmic equtions 5G Logrithmic grphs 5H Applictions of eponentil nd logrithmic functions REs of stud Grphs of = A k + C, where R +, for simple cses of, A, k, nd C R The grph of = log () s the grph of the inverse function of =, including the reltionships log ( ) =, log ( ) = Simple pplictions of eponentil functions of the bove form to model growth nd dec in popultions nd the phsicl world, pprecition nd deprecition of vlue in finnce; the interprettion of initil vlue, rte of growth nd dec, nd long-run vlues in these contets nd their reltionship to the prmeters A, k nd C Solution of equtions of the form A + c = d using ect or pproimte vlues on given domin nd interprettion of these equtions Inde lws nd logrithm lws, including their ppliction to the solution of simple eponentil equtions ebookplus Introduction Functions in which the independent vrible is n inde number re clled indicil or eponentil functions. For emple: f () = where > nd is n eponentil function. It cn be shown tht quntities which increse or decrese b constnt percentge in prticulr time cn be modelled b n eponentil function. Eponentil functions hve pplictions in science nd medicine (for emple, dec of rdioctive mteril, or growth of bcteri like those shown in the photo), nd finnce (for emple, compound interest nd reducing blnce lons). Digitl doc Quick Questions 6 Mths Quest Mthemticl Methods Cs for the Csio ClssPd
5 Inde lws Recll tht number,, which is multiplied b itself n times cn be represented in inde nottion. n = nlots of Inde (or power or eponent) Bse where is the bse number nd n is the inde (or power or eponent). n is red s to the power of n or to the n. Multipliction When multipling two numbers in inde form with the sme bse, dd the indices. For emple, = = 7 Division When dividing two numbers in inde form with the sme bse, subtrct the indices. For emple, = = 6 Rising to power To rise n indicil epression to power, multipl the indices. For emple, ( ) = = + + = = m n = m + n m n = m n ( m ) n = m n = mn Rising to the power of zero An number rised to the power of zero is equl to one. =, For emple = - = [] or = ( ) ( ) = 8 8 = So = [] Using [] nd [] we hve =. Products nd quotients Note the following. For emple, ( ) = ( ) ( ) ( ) ( ) = = = (b) n = n b n n b = b n n Worked Emple Simplif. b ( ) c () 5 b 6 9 b d 8 pm ( p ) m 6 pm 6 5 Think Collect plin numbers ( nd ) nd terms with the sme bse. Write = Chpter 5 Eponentil nd logrithmic functions 7
Simplif b multipling plin numbers nd dding powers with the sme bse. (Note: = ) b Remove the brcket b multipling the powers. (The power of the inside the brcket is.) = 8 5 b ( ) = 6 Convert to plin number () first = 6 nd collect terms with the sme bse. Simplif b dding powers with the = 5 sme bse. c Write the quotient s frction. c ( ) ( ) 5 6 9 b b b = 9b Remove the brcket b multipling the powers. Simplif b first cncelling plin numbers. Complete simplifiction b subtrcting powers with the sme bse. (Note: =.) d Epnd the brckets b rising ech term to the power of. Convert to 7 nd collect like pronumerls. Simplif b first reducing the plin numbers, nd the pronumerls b dding the indices for multipliction nd subtrcting the indices for division. d 8 pm ( p) m 6pm 6 5 5 6 = b 9b 5 6 = 7 b 5 6 b = 7b 8pm pm = 6pm Simplif the indices of ech bse. = 6p 5 m 6 6 5 = 8 7 p p m m 6pm = 6p 6 + m + 5 6 5 Worked Emple Simplif 6 b 6b Think 7 6 b b. Method : Technolog-free Write the epression. Chnge the division sign to multipliction nd replce the second term with its reciprocl (turn the second term upside down). Write/displ 6b 6b 7 6 b b 6b = b 6b 7 6 b 8 Mths Quest Mthemticl Methods CAS for the Csio ClssPd
Remove the brckets b multipling the powers. Collect plin numbers nd terms with the sme bse. 5 Cncel plin numbers nd ppl inde lws. 6b b = 6b 7 6 b 9 6 6 6 = 8 b 6 7 = b 9 + 97 6 + 66 6 Simplif. = 9 Method : Technolog-enbled This eqution cn lso be nswered using CAS clcultor. On the Min screen, tp: Action Trnsformtion simplif Complete the entr line s: 6 6 b b 7 6 b b Write the nswer. 6b 6b 7 6 b b = 9 Worked Emple Write in simplest inde nottion nd evlute. 6 b 95 7 Think Write Rewrite the bses in terms of their prime fctors. 6 = ( ) Simplif the brckets using inde nottion. = ( ) Remove the brckets b multipling the powers. = 8 Simplif b dding the powers. = 5 Evlute s bsic number. = 8 b Rewrite the bses in terms of their b 9 5 ( ) = prime fctors 7 ( ) 5 Chpter 5 Eponentil nd logrithmic functions 9
( ) Simplif the brckets using inde nottion. = ( ) Remove the brckets b multipling the = powers. 9 Write in simplest inde form. = 5 5 Evlute s bsic number. = 5 Comple epressions involving terms with different bses hve to be simplified b replcing ech bse with its prime fctors. Worked Emple n + Simplif n 8 6 n Think. Write Rewrite the bses in terms of their prime fctors. n 8 6n n+ n ( ) = ( ) n n+ n ( ) Simplif the brckets using inde nottion. = ( ) n n+ Remove the brckets b multipling powers. = n n Collect terms with the sme bse b dding the powers in the products nd subtrcting the powers in the quotients. n n+ n+ 5 Simplif. = 6n + - n + n + - n + = n + - n + = n + - n = n + n + -(n- ) n + -(n- ) REMEMBER. Inde lws: () m n = m + n (b) m n = m - n (c) = (d) ( m ) n = mn (e) (b) n = n b n n n (f) b = bn. To simplif indicil epressions: () when deling with questions in the form (epression ) (epression ), replce epression with its reciprocl nd chnge to (b) remove brckets using lws (d), (e) nd (f) (c) collect plin numbers nd terms of the sme bse (d) simplif using lws (), (b) nd (c). Mths Quest Mthemticl Methods CAS for the Csio ClssPd
Eercise 5A Inde lws WE, b Simplif. 5 b 5 5 7 (5 ) c () 5 d m p 5 (mp ) m p 6 WE c Simplif. 7 b 8 b 5 b b 9 () b c ( 5 ) 6 d p q ( pq ) WE d Simplif. c 6pm pm 9pm 5 8 7 6 uv9 ( u) v uv 6 5 WE Simplif. 5 b 9b 8 5 b b b d b ( ) 5 7 6 ( 5e ) f 8e f ef5 5kd 6kd ( k) 5( kd) 5 MC pm pm cn be simplified to: A p m B p m 6 C p m 8 D p m E b 6 6 5 cn be simplified to: 5 ( ) A 5 B 5 C 6 D c b b b 5 is equl to: A 8 B 6 C 6 b D 6 Simplif ech of the following. n 5 z z + n n n n 7 WE Write in simplest inde nottion. b 9 b6 5 ( nm+ ) 8 b 7 9 7 8 c 5 5 d 5 8 5 e 7 6 5 8 Write in simplest inde nottion nd evlute. 5 7 e ( 65 ) ( 5) 5 9 WE Simplif. b 9 5 7 f ( 5) ( 5) c g f ( 6 ) ( 5) 8 6 n 9 n+ b 6n 6 d 5n 6n 9 e* 5 n 5 n+ n+ 8n 6 n 5n+ + 5n *Hint: Fctorise the numertor nd denomintor first. E 5 E 6 n+ m n5 5m 8 5 c d h 7 ( ) 7 8 9 5 7 9 n+ n n n+ Chpter 5 Eponentil nd logrithmic functions
5B n+ MC In simplest inde nottion, 6 n 6 6n is equl to: A 6 n + 5 B 6 5n + C 6 n + 5 D 6 9 E 6 n + 9 negtive nd rtionl powers negtive powers Wherever possible, negtive inde numbers should be epressed s positive inde numbers using the simple rule: When n inde number is moved from the numertor to denomintor or vice vers, the sign of the power chnges. This is esil verified s follows: n =, n = since = n n = n using division rule for indices = n simplifing the inde. In other words, n n = nd = n n A simple w to remember this rule is chnge the level, chnge the sign. WoRkED EMPlE 5 Epress ech of the following with positive inde numbers. 5 8 b 5 ( ) ebookplus Tutoril int-9 Worked emple 5 ThInk Remove the brckets b rising the denomintor nd numertor to the power of. Interchnge the numertor nd denomintor, chnging the signs of the powers. Simplif b epressing s frction to the power of. WRITE 5 5 8 = 8 = 8 5 = 8 5 b Remove the brckets b multipling powers. b ( ) Collect terms with the sme bse b dding the powers on the numertor nd subtrcting the powers on the denomintor. 5 Rewrite the nswer with positive powers. = = 5 = = = 6 7 6( ) 7 Mths Quest Mthemticl Methods Cs for the Csio ClssPd
Rtionl powers Until now, the indices hve ll been integers. In theor, n inde cn be n number. We will confine ourselves to the cse of indices which re rtionl numbers (frctions). n, where n is positive integer, is defined s the nth root of : For emple, we know tht = but = = = Therefore, = Similrl, =, = etc. n is defined for ll nd n. In generl, + n = ( ) = ( ) = = m n n m m n m n Worked Emple 6 Evlute ech of the following without clcultor. 6 b 9 5 Think Write Rewrite the bse number in terms of its prime fctors. Remove the brckets b multipling the powers. 6 = ( ) = 6 Evlute s bsic number. = 6 b Rewrite the bse numbers of the frction in terms of their prime fctors. Remove the brckets b multipling the powers. Rewrite with positive powers b interchnging the numertor nd denomintor. Evlute the numertor nd denomintor s bsic numbers. b 9 5 = 5 = 5 = 5 = 5 7 Chpter 5 Eponentil nd logrithmic functions
Worked Emple 7 Simplif the following, epressing our nswer with positive indices. 7 8 6 b 6 5 Think Write 7 Write the epression. 8 6 Write using frctionl indices. = 87 6 Write 8 nd 6 in inde form. = ( 7) 7 ( 6) Multipl the powers. = 5 Simplif the powers. = 5 = b Write the epression. b 6 6 5 Epress the roots in inde nottion. = ( 6) ( 5) Remove the brckets b multipling the powers. Collect terms with the sme bse b subtrcting the powers. = = 5 Simplif the powers. = 6 Rewrite with positive powers. = 7 It is tempting to nswer this question using CAS clcultor. But the clcultor should be used wisel. To see wht hppens, On the Min screen complete the entr line s shown on the right. Then press E. 5 5 6 5 6 5 8 Note: The nswer provided b the clcultor would not be ccepted s it is not simplified sufficientl. In these cses, ou will need to work through the nswer lgebricll s shown erlier. Mths Quest Mthemticl Methods CAS for the Csio ClssPd
REMEMBER n n =, n = n m n n m n m n m = ( ) = ( ) = EERCIsE 5B negtive nd rtionl powers WE5 Epress ech of the following with positive inde numbers. 6 b 5 c WE5b Simplif ech of the following, epressing our nswer with positive inde numbers. ( ) ( ) ( ) ( m) ) m b ( ) c ( p ) p d 5 ( ) ( ) e ( ) ( 5 ) ( ) ( ) ) 5 ) f ( ) ( ) ( ) ebookplus Digitl doc SkillSHEET 5. Negtive nd rtionl powers WE6 Evlute the following without clcultor. 9 b 7 c 8 d 8 6 e 5 8 f 6 MC 5 5 simplifies to: 7 g 8 h 8 7 5 A 56 B 56 C 5 D 5 6 E 5 5 WE 7 Simplif ech of the following, epressing our nswer with positive indices. 9 8 b 6 c ( ) ( ) e 6 ( 6m ) m f g ( + ) + 6 d 8 5 h ( ) 5C Indicil equtions We cn solve equtions of the form = s follows: Tke the cube of both sides: ( ) = The left-hnd side becomes, so = 8. However, when the unknown (or vrible) is not bse number but is n inde number, different pproch is required. Chpter 5 Eponentil nd logrithmic functions 5
Worked Emple 8 Method : Ect solutions without clcultor To ttempt to solve inde equtions ectl, epress both sides of the eqution to the sme bse nd equte the powers. If m = n, then m = n. Find the vlue of in ech of the following equtions. = 8 b = 56 c 6 = 6 Think Write Write the eqution. = 8 Epress both sides to the sme bse. = Equte the powers. = b Write the eqution. b - = 56 Epress both sides to the sme bse. - = Equte the powers. - = Solve the liner eqution for b = 5 dding one to both sides. c Write the eqution. c 6 = 6 Epress both sides to the sme bse. 6 = (6 ) Remove the brckets b multipling the powers. 6 = (6) 6 Equte the powers. - = - 6 5 Subtrct from both sides to mke the subject. - = - 6 6 Add 6 to both sides to solve the eqution. = 5 More complicted equtions cn be solved using the sme technique. Worked Emple 9 Solve for n in the following eqution. n 6 n + = Think Write Write the eqution. n 6 n + = Epress both sides using the sme bse,. n ( ) n + = 5 Remove the brckets b multipling the powers. n n + = 5 Multipl the terms on the left-hnd side b dding the powers. 7n + = 5 5 Equte the powers. 7n + = 5 6 Solve the liner eqution for n. 7n = n = 7 6 Mths Quest Mthemticl Methods CAS for the Csio ClssPd
In some cses indicil equtions cn be epressed in qudrtic form nd solved using the Null Fctor Lw. Look for numbers in inde form similr to nd ppering in different terms. WoRkED EMPlE Solve for if 5 (5 ) 5 =. ThInk WRITE Write the eqution. 5 (5 ) 5 = Rewrite the eqution in qudrtic form. (5 ) (5 ) 5 = Note tht 5 = (5 ). Substitute for 5. Let = 5 Rewrite the eqution in terms of. 5 =. 5 Fctorise the left-hnd side. ( 5)( + ) = 6 Solve for using the Null Fctor Lw. = 5 or = 7 Substitute 5 for. 5 = 5 or 5 = 8 Equte the powers. 5 = 5 nd 5 = ebookplus Tutoril int-9 Worked emple 9 Stte the solution(s). = (5 = hs no solution.) WoRkED EMPlE Solve for given Note tht in step 9, the possible solution 5 = ws rejected becuse there is no vlue of for which it will be stisfied. Recll tht eponentil functions such s 5 re lws positive. Method : using Cs clcultor If nswers re not ect, the CAS clcultor cn be used to solve indicil equtions. + =. Write our nswer correct to deciml plces. 5 ThInk WRITE/DIsPl On the Min screen, tp: Action Advnced solve Complete the entr line s: Solve( + = 5, ) Check it is in deciml mode; then press E. Write the nswer. Solving + = for gives =.65, correct to 5 deciml plces. Chpter 5 Eponentil nd logrithmic functions 7
REMEMBER. If m = n, then m = n.. Inect solutions require the use of clcultor. Eercise 5C Indicil equtions WE8 Find the vlue of in ech of the following equtions. = b 5 = 65 c = d = e = 6 f 6 = 6 WE8b Find the vlue of n in ech of the following equtions. n + = 6 b 5 n + = 5 c n = 7 d 6 n + = e 9 5 n = f 6 n = 6 WE8c Find in ech of the following. = 8 - b 7 - = 9 + c 6 + = 8 d 9 + 5 = WE9 Solve for in ech of the following equtions: 8 = 6 b 5 5 = 5 c 7 + 8 = 8 d = 9 7 + 5 WE Solve for in ech of the following. ( ) + = b 6( ) + 8 = c ( ) = 5( ) d 5 (5 ) + 5 = 7 6 MC Consider the indicil eqution ( ) + 7 =. The eqution cn be solved b mking the substitution: = b = c = d = e = 7 MC The qudrtic eqution formed b the pproprite substitution in question 6 is: + 7 = b + 7 = c + + 7 = d + 7 = e 9 + = 8 MC The solutions to the eqution in question 7 re equls: or b or c or d or e or 9 WE Solve for. Write our nswer correct to deciml plces. = b = 8 c = 5 d = MC The nerest solution to the eqution = is: =.5 b =. c =.9 d = e =. 5D Grphs of eponentil functions Functions of the form f () =, where is positive rel number other thn nd is rel number, re clled eponentil functions. 8 Mths Quest Mthemticl Methods CAS for the Csio ClssPd
In generl, there re two bsic shpes for eponentil grphs: =, > or =, < < =, > Incresing eponentil Asmptote = =, < < Asmptote = Decresing eponentil However, in both cses: the -intercept is (, ) the smptote is = (-is) the domin is R the rnge is R +. Verif the shpes of these grphs b grphing, s =, =, = nd = on grphics clcultor. Wht is the effect of chnging on the steepness of the grph? Reflections of eponentil functions The grph of = is obtined b The grph of = is obtined b reflecting = in the -is. reflecting = in the -is. =, > =, > Asmptote = =, > =, > Horizontl trnsltions of eponentil functions The grph of = + b is obtined b trnslting = :. b units to the right if b <. b units to the left if b >. For emple, the grph of = is obtined b trnslting = to the right units. Check this grph using grphics clcultor. Note lso tht = ( )( ) = 8 so tht the effect is identicl to tht of multipling b constnt. Verticl trnsltions of eponentil functions The grph of = + c is obtined b trnslting = :. up b c units if c >. down b c units if c <. Furthermore the eqution of the smptote becomes = C. For emple, the grph of = 5 is obtined b trnslting = down b 5 units. The eqution of the smptote is = 5. The -intercept is. Check this grph using grphics clcultor. 5 units = = 5 5 Asmptote = units Asmptote = = = 5 (Asmptote = 5) Chpter 5 Eponentil nd logrithmic functions 9
Diltion from the -is The grph of = A (for positive, rel vlues of A) hs diltion fctor of A. The grph is stretched long the -is, w from the -is (s ech -vlue is being multiplied b the constnt A). Consider the grphs below. The -intercept in ech cse is equl to A. Also, s A increses, the grph becomes steeper; s A decreses, the grph becomes less steep. The domin, rnge nd smptotes re the sme s for f () =. (, ) f() = f() = f() = (, ) (, ) Asmptote = (, ) (, ) (, ) (, ) f() = (, ) (, ) f() = f() = Asmptote = Diltion from the -is The grph of = k (for k > ) hs diltion fctor of k from the -is. The grph is sid to be stretched long the -is. Consider the grphs t right. The -intercept is (, ) in ech cse. As k increses, the grph becomes steeper nd closer to the -is. The domin, rnge nd smptotes re the sme s for f () =. (, ) f() = f() = f() = Asmptote = Worked Emple Find the eqution of the smptote nd the -intercept for ech of the following. Hence, sketch the grph of ech nd stte its domin nd rnge. f : R R, f () = + 5 b f: R R, f( ) = Think Write/drw Write the rule. f () = + 5 The grph is the sme s = trnslted units left nd 5 units down. Stte the smptote. The smptote is = 5. Evlute when = to find the -intercept. When =, = 5 = Therefore, the -intercept is (, ). 5 Locte the -intercept nd smptote on set of es. 6 Sketch the grph of the eponentil function using the -intercept nd smptote s guide. 5 f() = + 5 Asmptote = 5 Mths Quest Mthemticl Methods CAS for the Csio ClssPd
7 Use the grph to stte the domin nd rnge. The domin is R nd the rnge is ( 5, ). b Write the rule. b f( )= Find the -intercept b letting = or recll tht the -intercept is equl to A in f () = A. f ( ) = = = = The -intercept is. Locte the horizontl smptote. The horizontl smptote is the -is. Locte nother point on the grph. This is necessr to be ble to see the effect of the diltion. Locte the second point b substituting vlue for into the eqution nd evluting corresponding vlue. 5 Sketch the grph. Note: The smptote remins t = s there is no verticl trnsltion. f ( ) = = = = 6 Another point is (, 6) 6 f() = (, 6) (, ) Asmptote = 6 Stte the domin nd the rnge. The domin is R nd the rnge is R +. Worked Emple Use CAS clcultor to solve = 5 (correct to deciml plces) b finding the intersection of two grphs. Think Write/displ On the Grph & Tb screen, complete the function entr lines s: = = 5 Tick the nd nd tp!. Chpter 5 Eponentil nd logrithmic functions
To locte the point of intersection, tp: Anlsis G-Solve Intersect Write the nswer. = 5 when =.9, correct to deciml plces. REMEMBER. Generl shpes of grphs of eponentil functions: f () =, > ; f () =, < < f() =, > In both cses, the -intercept is (, ), the Asmptote = smptote is =, the domin = R, nd the rnge = R +.. Reflections: f () =, > ; f () = -, > ; f () =, > ; f () = -, > Asmptote = f() =, > f() =, > Asmptote = f() =, < < f() =, > f() =, > Asmptote =. Trnsltions f () =, > ; f () = +, f () =, > ; f () = + c, >, b > >, c > f() = +b, >, b > f() = + c, >, c > Asmptote = b b f() =, > = c c c f() =, > Mths Quest Mthemticl Methods CAS for the Csio ClssPd
. Diltions f () =, > ; f () = A, f () =, > ; f () = k, A >, > >, k > f() = A, A >, > f() =, > f() = f() = f() = Asmptote = A (, ) Asmptote = EERCIsE 5D ebookplus Digitl doc SkillSHEET 5. Substitution in eponentil functions Grphs of eponentil functions Sketch the grph of ech of the following on seprte es. (Use tble of vlues or cop CAS clcultor screen). = b = 5 c = d = e = f = g = h =.5 Sketch the following grphs, using tble of vlues or b coping CAS clcultor screen. Stte the eqution of the smptote nd the -intercept for ech. = ( ) b =.5( ) c = ( ) d = WE Find the eqution of the smptote nd the -intercept for ech of the following. Hence, sketch the grph of ech nd stte its domin nd rnge. f : R R, f () = b f : R R, f () = + c f : R R, f () = 5 d f : R R, f () = + e f : R R, f () = f f : R R, f () = + g f : R R, f () = 6 + h f : R R, f () = + 5 MC The rule for the grph t right is: A = B = C = D = + E = Asmptote = b The rule for the grph t right is: A = B = C = + D = + E = (, ) = Chpter 5 Eponentil nd logrithmic functions
ebookplus Digitl doc WorkSHEET 5. 5E 5 WEb Sketch grph of ech of the following, stting the domin nd rnge. f : R R, f () = b f : R R, f () = c f : R R, f () = d f : R R, f () = 5 e f : R R, f( ) = 5 f f : R R, f( ) = 6 Sketch grph of f () = +, stting the domin nd rnge. Compre our nswer to tht found using CAS clcultor. 7 WE Use CAS clcultor to solve the following indicil equtions using the intersection of two grphs. Give nswers rounded to deciml plces. = b = c = d = + e = + logrithms The inde, power or eponent () in the indicil eqution = is lso known s logrithm. This mens tht = cn be written in n lterntive form: = log () =, which is red s the logrithm of to the bse Bse numerl is equl to. For emple, = 9 cn be written s log (9) =. 5 = cn be written s log () = 5. In generl, for > nd : = is equivlent to = log (). Using the indicil equivlent, it is possible to find the ect vlue of some logrithms. Bse Logrithm WoRkED EMPlE Evlute the following without clcultor. log 6 (6) b log ( ) 8 ThInk WRITE Let equl the quntit we wish to find. Let = log 6 (6) Epress the logrithmic eqution s n indicil eqution. Epress both sides of the eqution to the sme bse. 6 = 6 6 = 6 Equte the powers. = b Write the logrithm s logrithmic eqution. b Let = Epress the logrithmic eqution s n indicil eqution. Epress both sides of the eqution to the sme bse. = = log 8 8 = Equte the powers. = = ( ) Mths Quest Mthemticl Methods Cs for the Csio ClssPd
Worked Emple 5 Logrithm lws The inde lws cn be used to estblish corresponding rules for clcultions involving logrithms. These rules re summrised in the following tble. Nme Rule Restrictions Logrithm of product log (mn) = log (m) + log (n) m, n > >, Logrithm of quotient log m n = log (m) - log (n) m, n > > nd Logrithm of power log (m) n = n log (m) m > > nd Logrithm of the bse log () = > nd Logrithm of one log () = > nd It is importnt to remember tht ech rule works onl if the bse,, is the sme for ech term. Note tht it is the logrithm of product nd logrithm of quotient rules tht formed the bsis for the pre-97s clcultion device for multipliction nd division the slide rule. Simplif, nd evlute where possible, ech of the following without clcultor. log (5) + log () b log () + log (8) - log () Think Write Appl the logrithm of product rule. log (5) + log () = log (5 ) Simplif. = log () b Multipl the bse numerls of the logs being dded since their bses re the sme. Appl the logrithm of quotient lw. Simplif, noting tht is power of. Evlute using the logrithm of power nd logrithm of the bse lws. b log () + log (8) log () = log ( 8) log () = log (96) log () = log (96 ) = log () = log () 5 = 5 log () = 5 Worked Emple 6 Simplif log (5) log (). Think Write Epress both terms s logrithms of inde numbers. log (5) log () = log (5) log () Simplif ech logrithm. = log (5) log () Appl the logrithm of quotient lw. = log (5 ) Simplif. = log or log (. ) 5 5 Chpter 5 Eponentil nd logrithmic functions 5
Worked Emple 7 Simplif ech of the following. log 8 ( 9) b log () + c 5 log () - log ( ) 8 Think Write Epress ech bse numerl s powers to the sme bse, 7. log 8 ( 9) log 8 ( 7) = log ( ) log ( 7) Appl the logrithm of power lw. = log 8 ( 7 ) log ( 7) Simplif b cncelling out the common fctor of log 8 (7). b Epress log () s log () nd s logrithm to bse lso. 8 = 8 b log () + = log () + log () Simplif using the logrithm of product lw. = log () c Epress 5 log () s log () 5 nd s log (). c 5 log () - = log () 5 log () Epress log () s log (). = log () 5 log () 8 Simplif using the logrithm of quotient lw. = log = log 5 5 Worked Emple 8 Evlute ech of the following epressions, correct to deciml plces. log (5) b log 7 (8) Think & b On the Min screen, complete the entr line s: log (5) log 7 (8) Press E fter ech entr. Note: The log templte is under the ) tb. Write/displ Write the nswer. log (5) =. b log 7 (8) =.69, correct to deciml plces. 6 Mths Quest Mthemticl Methods CAS for the Csio ClssPd
REMEMBER. If = then log ( ) =, where = the bse, = the power, inde or logrithm nd = the bse numerl. Note tht >,, nd therefore >.. Log lws: () log (m) + log (n) = log (mn) (d) log () = m (b) log ( m) log ( n) = log n (e) log () = (c) log (m n ) = n log (m) Eercise 5E Logrithms Epress the following indicil equtions in logrithmic form. = 8 b 5 = c 5 = d. = - e b n = f = Epress the following logrithmic equtions in indicil form. 6 log (6) = b log ( ) = 6 c log = d log (7) = e log 5 (65) = f log (8) = 7 9 g log = - h log b () = MC The vlue of log 5 (5) is: b 5 c d e MC When epressed in logrithmic form, 8 = 5 is: log (8) = 5 b log (5) = 8 C log 8 (5) = D log 5 () = 8 E log 8 () = 5 5 MC When epressed in indicil form, log ( ) = is: = B = C = D = E = 6 WE Evlute ech of the following. log (6) b log (8) c log 5 (5) d log e log () f log (. ) g log (.5) h log 6 i log () j log k log ( ) l log n (n 5 ) 7 WE 5 Simplif, nd evlute where possible. log (8) + log () b log (7) + log (5) c log () + log (5) d log 6 (8) + log 6 (7) e log () log (5) f log (6) log () 9 5 g log + log ( ) h log ( 5 ) + log 8 WE 6 Simplif ech of the following. i log (8) log () + log (5) log (5) + log () b log (8) + log () c log 5 () log 5 () d log () log (8) e log (7) log (6) f log ( ) + log () g log (6) + log () h log ( + ) log ( ) Chpter 5 Eponentil nd logrithmic functions 7
5F 9 WE 7 Simplif the following. log ( 5) log ( 8) b log ( 5) log ( 9) log ( 6) log ( ) e f log ( ) log ( ) MC The epression log () is equl to: c g log ( 8) log ( 6) log ( + ) log ( + ) log () log () B log () log () C D log () E log () + log () d log ( ) log ( ) log 5 ( 7) log ( 9) MC The epression log 5 () is equl to: log 5 () B log 5 () C 5 log () D log 5 () + log 5 () E 5 MC The epression log (6) + log (5) cn be simplified to: 6 log () B C log 5 D log () E log MC The epression log ( 5 ) log ( cn be simplified to: ) log ( ) B log ( ) 5 C 5 D log ( 5 ) E log ( 7 ) WE 7b Epress ech of the following in simplest form. log (7) + b log (6) + c log 5 () d + log () e log () log (6) + f 5 + log ( ) 5 WE 8 Evlute, correct to deciml plces where pproprite. log (6) b log (8) c log (.) d log (9) e log () + log (7) Solving logrithmic equtions Logrithms to the bse Logrithms to the bse re clled common logrithms nd cn be evluted using the log function on clcultor. Note: The logrithm of negtive number or zero is not defined. Therefore: log () is defined for >, if > This cn be seen more clerl using inde nottion s follows: Let n = log (.) Therefore, n = (indicil equivlent of logrithmic epression). However, n > for ll vlues of n if > (positive bsed eponentils re lws positive). Therefore, >. Worked Emple 9 Find if log (9) =. Think Write Write the eqution. log (9) = Simplif the logrithm using the logrithm of power lw nd the fct tht log () =. log ( ) = log () = = Solve for b dding to both sides. = 8 Mths Quest Mthemticl Methods CAS for the Csio ClssPd
WoRkED EMPlE Solve for if log 6 () =. ThInk WRITE Write the eqution. log 6 () = Epress in inde form. Therefore, = 6. Evlute the inde number. = = 6 6 WoRkED EMPlE Solve for given tht log (5) =, >. ThInk WRITE Write the eqution. log (5) = Divide both sides b. log (5) = Write s n inde eqution. Therefore, = 5. Epress both sides of the eqution to the sme bse, 5. = 5 5 Equte the bses. Note tht = 5 is rejected s solution, becuse >. = 5 WoRkED EMPlE Solve for correct to deciml plces, if = 7. ThInk WRITE Write the eqution. = 7 Tke log of both sides. log ( ) = log (7) Use the logrithm of power lw to bring the power,, to the front of the logrithmic eqution. log () = log (7) Divide both sides b log () to get b itself. Therefore = log ( 7 ) log ( ) 5 Evlute the logrithms correct to deciml plces, =. 85 t lest one more thn the nswer requires.. 6 Solve for. =.88 Therefore, we cn stte the following rule: If log n ( b = b, then = log ( n ) n ) ) = log ( b Chpter 5 Eponentil nd logrithmic functions 9
This rule pplies to n bse n, but is the most commonl used bse for this solution technique. REMEMBER. Logrithmic equtions re solved more esil b: () simplifing using log lws (b) epressing in inde form (c) solving s required. If log n ( b) = b, then = = log ( b). log ( ) n Eercise 5f Solving logrithmic equtions WE 9 Find in ech of the following. log () = b log 9 () = c log = 9 d log = e log (8) = f log (8) = g log () = WE Solve for. log () = b log () = c log 5 () = d log () = e log 8 () = - f log () = - g log ( - ) = h log ( + ) = i log () = j log 6 () = k log (5) - log () = log () - log (8) WE Solve for given tht: log (6) = b log (5) = c log (6) = 6 d = log g log + (7) = e log ( 6) = f 5 log (65) = h log = MC The solution to the eqution log 7 () = is: = b = c = d = e = b If log 8 () =, then is equl to: 96 b 5 c 6 d e c Given tht log ( ) =, must be equl to: b 6 c 8 d e 9 d The solution to the eqution log () - = log ( - 8) is: = 8 b = 6 c = 9 d = e = 5 WE Solve the following equtions for, correct to deciml plces. = b =.6 c =.7 d 5 = 8 e.7 = f = 8 g + = 5 h = 7 i 8 =.75 5 5 Mths Quest Mthemticl Methods CAS for the Csio ClssPd
ebookplus Digitl doc WorkSHEET 5. 5G 6 MC The nerest solution to the eqution = 5 is: A =.86 B =. C =.5 D = E =.5 7 MC The nerest solution to the eqution.6 = is: A =.8 B =. C =.8 D =.7 E =. logrithmic grphs The grphs of = log () nd = re reflections of ech other cross the line =. Functions such s these tht re reflections of ech other in the line = re clled inverses of ech other. Consider the logrithm log ( ). This logrithm cn be simplified using the log lws. log ( ) = log () = = Notice how the logrithm with bse nd the eponentil with bse hve cncelling effect on one nother, demonstrting tht the re inverse opertions. This is similr to the w tht multipliction nd division hve cncelling effect. Multipliction nd division re lso inverse opertions of ech other. Consider now the eponentil log (). As the logrithm with bse is the inverse opertion to the eponentil with bse, the epression log () simplifies to give. Tht is, log () =. The inverse properties of logrithms nd eponentils cn be used to plot the grphs of logrithmic functions. Alterntivel, tble of vlues cn be used. = log () undefined undefined..77.6 The grph of = log () does not eist for vlues of. It is n incresing function. There is verticl smptote long the -is, nd so there re no -intercepts. The -intercept for ll vlues of is lws (, ). Tht s becuse log () =. Another point on the grph is (, ). Tht s becuse log () =. The domin of the function is R + nd the rnge is R. Asmptote = ebookplus Interctivit int-6 Investigting logrithmic grphs f() = log () WoRkED EMPlE Sketch the grph of f () = log (). ThInk WRITE/DRW Relise tht f () = log () is the inverse of f () =, so these two grphs re reflections of ech other cross the line =. Alterntivel, recll the bsic shpe of the logrithmic grph. Sketch the bsic shpe on set of es. Mrk the -intercept (, ). Mrk second point on the grph (, ), which in this cse is (, ). f() = (, ) (, ) = f() = log () Chpter 5 Eponentil nd logrithmic functions 5
Worked Emple Find the eqution of the inverse of f () =. Think Write/displ On the Min screen, tp: Action Commnd Define Complete the entr line s: Define f () = Then press E. To swp the - nd -vlues nd to find the eqution of the inverse function, complete the entr line s: Solve(f () =, ) Then press E. Simplif the nswer using the chnge of bse formul. Write the nswer. f log e ( ) ( ) = for > log ( ) e f ( ) = log ( ) for > REMEMBER. The functions f () = log () nd f () = re inverse functions. The grphs of these functions re reflections of ech other cross the line =.. Tking the logrithm with bse is the inverse opertion to tking the eponentil with bse.. log ( ) =. log ( ) = Eercise 5G Logrithmic grphs Simplif the following epressions. log ( ) b log ( b) c log 5 (5 ) d 5 log ( ) 5 Mths Quest Mthemticl Methods CAS for the Csio ClssPd
5h WE Sketch the grphs of ech of the following. f () = log () b f () = log 5 () c f () = log 8 () d f () = log () e f () = log () f f () = log 5 () Compre the steepness of ech of the grphs in question, nd hence eplin how chnging the bse,, ffects the steepness of logrithmic grph of the tpe f () = log (). WE Find the eqution of the inverse of the following. f () = b f () = c f () = log 5 () Further work on logrithmic grphs is vilble on the ebookplus. pplictions of eponentil nd logrithmic functions Eponentil nd logrithmic functions cn be used to model mn prcticl situtions in science, medicine, engineering nd economics. ebookplus Digitl docs Spredsheet Logrithmic grphs Etension Logrithmic grphs WoRkED EMPlE 5 A squre sheet of pper which is. mm thick is repetedl folded in hlf. Find rule which gives the thickness, T mm, s function of the number of folds, n. b Wht is the thickness fter folds? c How mn folds re required for the thickness to rech 6 cm? ThInk T =. when n = nd doubles with ech fold. This doubling implies tht the bse should be. Complete tble of vlues showing the thickness, T, for vlues of n from to 5. WRITE When n =, T =. nd s n increses b, T doubles. n 5 T....8.6. Determine the rule for T(n). There is doubling term ( n ) nd multipling constnt for the strting thickness (.). Compre the rule for T(n) ginst the tble of vlues in step. T(n) =.( n ) b Substitute n = into the formul for T. b When n =, T() =.( ) Clculte T. T =. mm c Chnge 6 cm to millimetres. c 6 cm = 6 mm Substitute T = 6 into the formul. When T = 6, 6 =. ( n ) Divide both sides b.. 6 = n Tke log of both sides. log (6) = log ( n ) 5 Use the logrithm of power lw to bring the log (6) = n log () power n to the front of the logrithm. 6 Divide both sides b log (). n = log ( 6 ) log ( ) Chpter 5 Eponentil nd logrithmic functions 5
7 Evlute. n 9. 8 Round the nswer up to the nerest whole number since the number of folds re positive integers nd if ou round down the thickness will not hve reched 6 mm. Therefore, n = folds. WoRkED EMPlE 6 The price of gold since 98, P (dollrs per ounce), cn be modelled b the function P = + 5 log (5t + ), where t is the number of ers since 98. Find the price of gold per ounce in 98. b Find the price of gold in 6. c In wht er will the price pss $55 per ounce? ebookplus Tutoril int-9 Worked emple 6 ThInk WRITE Stte the modelling function. P = + 5 log (5t + ) Determine the vlue of t represented b the er 98. In 98, when t =, Substitute t into the modelling function. P = + 5 log [5() + ] = + 5 log () Evlute P. P = b Repet prt b determining the vlue of t represented b the er 6. Substitute the vlue of t into the modelling function nd evlute P. c Since P = 55, substitute into the modelling function nd solve for t. Simplif b isolting the logrithm prt of the eqution. Epress this eqution in its equivlent indicil form. b t = 6 98 = 6 When t = 6, P = + 5 log [5(6) + ] = + 5 log () = + 5.86 = $55.86 c 55 = + 5 log (5t + ) 5 = 5 log (5t + ) = log (5t + ) = 5t + Solve this eqution for t. = 5t + 999 = 5t 99.8 = t 5 Convert the result into ers. The price will rech $55 in 99.8 ers fter 98. The price of gold will rech $55 in 98 + 99.8 = 8 (pproimtel). 5 Mths Quest Mthemticl Methods Cs for the Csio ClssPd
REMEMBER.. Red the question crefull. Use the skills developed in the previous sections to nswer the question being sked. Eercise 5h Applictions of eponentil nd logrithmic functions WE5 Prior to mice plgue which lsts 6 months, the popultion of mice in countr region is estimted to be. The mice popultion doubles ever month during the plgue. If P represents the mice popultion nd t is the number of months fter the plgue strts: epress P s function of t b find the popultion fter: i months ii 6 months c clculte how long it tkes the popultion to rech during the plgue. WE6 The popultion of town, N, is modelled b the function N = 5 (.t ) where t is the number of ers since 98. Find the popultion in 98. b Find the popultion in: i 985 ii 99. c Wht is the predicted popultion in 5? d In wht er will the popultion rech? The weight of bb, W kg, t weeks fter birth cn be modelled b W = log (8t + ). Find the initil weight. b Find the weight fter: i week ii 5 weeks iii weeks. c Sketch the grph. d When will the bb rech weight of 7 kg? If $A is the mount n investment of $P grows to fter n ers t 5% p..: write A s function of P b use the function from to find the vlue of $ fter ers c clculte how mn ers it will be until n investment of $ reches $6 5. 5 The vlue of cr, $V, decreses ccording to the function V = 5 5.t, where t is the number of ers since the cr ws purchsed. Find the vlue of the cr when new. b Find the vlue of the cr fter 6 ers. c In how mn ers will the cr be worth $? 6 The temperture, T ( C), of cooling cup of coffee in room of temperture C cn be modelled b T = 9( 5. t ), where t is the number of minutes fter it is poured. Find the initil temperture. b Find the temperture: i minutes fter pouring ii 6 minutes fter pouring. c How long is it until the temperture reches hlf its initil vlue? 7 A number of deer, N, re introduced to reserve nd its popultion cn be predicted b the model N = (. t ), where t is the number of ers since introduction. Find the initil number of deer in the reserve. b Find the number of deer fter: i ers ii ers iii 6 ers. Chpter 5 Eponentil nd logrithmic functions 55
c How long does it tke the popultion to treble? d Sketch the grph of N versus t. e Eplin wh the model is not relible for n indefinite time period. 8 After reccling progrm is introduced, the weight of rubbish disposed of b household ech week is given b W = 8(. 5 ), where W is the weight in kg nd t is the number of weeks since reccling ws introduced. Find the weight of rubbish disposed of before reccling strts. b Find the weight of rubbish disposed of fter reccling hs been introduced for: i weeks ii weeks. c How long is it fter reccling strts until the weight of rubbish disposed of is hlf its initil vlue? d i Will the model be relistic in ers time? ii Eplin. 9 The number of hectres (N) of forest lnd destroed b fire t hours fter it strted, is given b N = log (5t + ). Find the mount of lnd destroed fter: i hour ii hours iii hours. b How long does the fire tke to burn out 55 hectres? A discus thrower competes t severl competitions during the er. The best distnce, d metres, tht he chieves t ech consecutive competition is modelled b d = 5 + log (5n), where n is the competition number. Find the distnce thrown t the: i st ii rd iii 6th iv th competition. b Sketch the grph of d versus n. c How mn competitions does it tke for the thrower to rech distnce of 5 metres? The popultion, P, of certin fish t months fter being introduced to reservoir is P = (.8t ), t. After months, fishing is llowed nd the popultion is then modelled b P = 5 + 9 log [(t - 9)], t. Find the initil popultion. b Find the popultion fter: i 5 months ii 5 months iii 5 months iv months. c How long does it tke the popultion to pss? A bll is dropped from height of 5 metres nd rebounds of its previous height. to 7 Find the rule tht describes the height of the bll (h metres) fter n bounces. b Find the height fter: i bounces ii 8 bounces. c Sketch the grph of the height of the bll fter n bounces. A computer pprecites in vlue b % per er. If the computer costs $5 when new, find: the rule describing the vlue, V, of the computer t n time, t ers, fter purchse. b the vlue of the computer fter 6 ers. c the number of ers it tkes to rech double its originl vlue. 56 Mths Quest Mthemticl Methods CAS for the Csio ClssPd
From the strt of 996, smll mining town hs seen sted increse in popultion until s the price of minerls improved nd mining ws etended. Yer 996 997 998 999 Popultion (P) 7 75 8 87 95 Let 996 be t = ; then 997 would be t = etc. Plot P ginst t. b Wht does the shpe of the curve look like? c Clculte the rtio of the popultion in 997 to the popultion in 996. d Clculte ll the rtios in successive ers, nd hence estimte the percentge nnul increse or growth. To obtin n ccurte estimte of popultion growth, follow these steps: e On the tble bove evlute log (P). f Plot log (P) ginst t. Are the points pproimtel colliner? g Drw line of best fit nd find its grdient nd the intercept on the -is. h Write the eqution for the line. i Show tht P = 7(.7) t. Is this close to our estimte in d? j Use this formul to estimte the popultion in nd. k When might the popultion hve reched? l In fct there ws downturn in the popultion s the mine output decresed. From onwrds there ws n nnul decline of % in popultion. During which er did the popultion reduce to below 6? 5 A used cr delership keeps dt on the vlue of the Frud Atls (new t the strt of 5) over 5 ers. Yer (t) 5 6 7 8 9 Vlue (V) 5 5 5 8 5 8 Let t = 5 be = ; then 6 would be = etc. Plot V ginst. b Wht does the shpe of the curve look like? c Clculte the rtios of vlues of the cr in successive ers, e.g. V 6 V 5. etc. d Estimte the nnul rte of deprecition. e For more ccurte result, evlute log (V). f Plot log (V) ginst. Are the points pproimtel colliner? g Drw line of best fit nd find its grdient nd the intercept on the -is. h Write the eqution for the line. i Show tht V = 5 (.79) (or formul close to it). j Wht is the nnul rte of deprecition? k Use this formul to estimte the cr s vlues in nd. l When will it rech vlue of $7? 6 Johnnes Kepler ws Germn stronomer born in the 6th centur. He used dt collected b Tcho Brhe to formulte n eqution or lw connecting the period of plnet s revolution round the sun to the rdius of its orbit. The following tble contins the dt Kepler used. The rdius of the orbit is epressed s proportion of Erth s orbit (with the period given in ds). Plnet Rdius (R) of orbit Period (T) Mercur.89 87.77 Venus.7.7 Erth. 65.5 Mrs.5 686.98 Jupiter 5..6 Chpter 5 Eponentil nd logrithmic functions 57
Plot T ginst R (using (,) too). Wht does the grph look like? The grph hs the form T = R b, where nd b re constnts. b To find them, find log (R) nd log (T). c Plot log (T) ginst log (R) on grph pper. Are the points colliner? d Drw line of best fit, nd find its grdient, correct to deciml plces. e Red off the intercept on the -is, nd write it s the equivlent logrithm. f Write n eqution for the stright line. g B trnsposition show tht T = 65.5R.5 (or formul close to it) h If Sturn s orbit hs rdius of 9.5, find its period using the formul bove. The ctul period is 759. ds. Wh is there difference in the results? i Kepler s Lw is T = kr. Wht is the vlue of k? 58 Mths Quest Mthemticl Methods CAS for the Csio ClssPd
Summr Inde lws m n = m + n m n = m n ( m ) n = mn = ( b) n = n b n n n b = bn Negtive nd rtionl powers n n =, n = n m n n m n m n m = ( ) = ( ) = Indicil equtions If m = n, then m = n. A grphics clcultor m be used to solve indicil equtions, using the solve function. Grphs of eponentil functions f( ) =, > ; f ( ) =, < < -intercept is (, ) Asmptote is = (-is) Domin = R Rnge = R + Reflections: f ( ) =, > ; f ( ) =, > f() =, > Asmptote = f() =, < < Asmptote = f() =, > f ( ) =, > ; f ( ) =, > f() =, > Asmptote = f() =, > f() =, > Asmptote = Chpter 5 Eponentil nd logrithmic functions 59
Trnsltions f ( ) =, > ; f ( ) = + b, >, b > f() = +b, >, b > b f() =, > b Asmptote = f ( ) =, > ; f ( ) = + c, >, c > f() = + C, >, C > Asmptote = C C C f() =, > Diltions f ( ) =, > ; f ( ) = A, A >, > f() = A, A >, > f() =, > f ( ) =, > ; f ( ) = k, >, k > A Asmptote = f() = f() = f() = (, ) Asmptote = Logrithms If =, then log ( ) =, where = the bse, = the power, inde or logrithm, nd = the bse numerl. Log lws: log (m) + log (n) = log (mn) m, n > log (m) - log (n) = log log (m n ) = n log (m) m > log () = log () = m n m, n > 6 Mths Quest Mthemticl Methods CAS for the Csio ClssPd
Solving logrithmic equtions Logrithmic equtions re solved more esil b:. simplifing using log lws. epressing in inde form. solving s required. If log ( b) = b, then = = log ( b) log ( ) log ( ) = log ( ) = Logrithmic grphs The logrithmic function f ( ) = log () is the inverse function of the eponentil function f ( ) =. f() =, > = f() = log (), > -intercept is (, ) Asmptote is = Domin = R + Rnge = R Chpter 5 Eponentil nd logrithmic functions 6
ChPTER REVIEW short nswer Simplif the following epression with positive indices. ( 6 6 ) ( 7 9) Solve the following equtions. = 5 b 8 + = Find the solutions to: 9 ( ) 6 = b + + = For the function with the rule f ( ) = + : find the -intercept b stte the eqution of the smptote c sketch the grph of f ( ) d stte the domin nd rnge. 5 For the function with the rule f ( ) = : find the -intercept b stte the eqution of the horizontl smptote c find second point on the grph d sketch the grph of f ( ) e stte the domin nd the rnge of the function. 6 Evlute log 7 b Epress in terms of if log ( ) + log ( ) = log ( + ). 7 Simplif the following. log (5) log (6) b log 5 ( log ( ) 8 Solve ech of the following. log 6 () = b log (5) = 6 c log ( + 6) log (5) = 9 If = log e (7 6) +, then =? 5 5 ) If f () = : sketch the grph of f () nd lbel: i the -intercept ii the eqution of the smptote b sketch the line =, use this line to sketch the inverse function g() = log (), nd lbel: i the -intercept ii the eqution of the smptote. The number of bcteri in culture, N, is given b the eponentil function N = 5(.t ), where t is the number of ds. Find the initil number of bcteri in the culture. b Find the number of bcteri (to the nerest ) fter: i 5 ds ii ds. c How mn ds does it tke for the number of bcteri to rech? The grph of the function f with rule f ( ) = log e ( + ) + intersects with the es t the points (, ) nd (, b). Find the ect vlues of nd b. b Hence, sketch the grph of the function with rule f ( ) = log e ( + ) + on the es below. Lbel n smptote with its eqution. MulTIPlE ChoICE [VCAA ] When simplified, ( ) 5 is equl to: 7 A 7 B 7 C 7 7 7 D 6 E 5 6 m p 5m p ( ) m be simplified to: m p m7 p m A m m 7p6 B p C 5 p D m p 5 9 The vlue of 5 6 A D 5 E 5 m p 5 is: B 5 C 5 E 5 If 5 = 5, then is equl to: A B D E 5 C 6 6 Mths Quest Mthemticl Methods Cs for the Csio ClssPd
5 If 7( ) + 6 =, then is equl to: or 6 b or c or 8 d or e or 6 The rule for the grph below could be: = b = + c = d = e = + Questions 7 to 9 refer to the function defned b the rule = +. 7 The grph which best represents this function is: B c E 7 d 8 The domin is: (, ) b [ -, ) c R + d R e R\{ - } 9 The rnge is: [ -, ) b R c R + d (, ) e ( -, ) When epressed in log form, 5 = 5 becomes: log (5) = 5 b log 5 () = 5 c log 5 (5) = d log (5) = 5 e log 5 () = 5 The vlue of log 7 (9) + log (8) is: b 7 c d 69 e The vlue of log 5 is nerest to: log 5 b 5 c d 9 e The epression log 7 ( ) simplifies to: log ( ) 7 5 5 b log ( ) c log ( ) d e 5 8 The solution to log 5 () = is: 5 b 5 c d 65 e 5 The vlue of if log () = 6 is: b 7 c 5 d e 7 6 If log ( ) + log () =, then is equl to: b c d 5 e 7 The solution to the eqution = is nerest to: = - b =.5 c =.6 d =. e = 8 The eqution of the grph shown below is: 7 5 (, ) = log 6 () b = log () c = log 8 () d = log () e = log () 9 The inverse of the grph below would be: A (, ) (, ) 7 Chpter 5 Eponentil nd logrithmic functions 6
B B C (, ) c D (, ) E [VCAA ] D If log e () - log e ( + ) = + log e (), then is equl to: A B ( + ) + C + (, ) D + E e ( + ) [VCAA ] e (, ) If k nd P re positive rel numbers, which one of the following grphs is most likel to be the grph of the function with eqution = e k + P? A If 5e =, then is equl to: A. log e () B log e (.) C D log e ( ) log ( 5) e E log (. ) e log e ( ) log ( 5 ) [VCAA ] Prts of the grphs of the functions with equtions = log e ( + ) nd = - re shown below. The solution to the eqution = log e ( + ) = - is closest to:. b.5 c.55 d.56 e.57 [VCAA ] The solution of the eqution e = is closest to: -.6 b -.5 c. d. e.575 [VCAA 5] 5 The best pproimte vlue for log 5 (6) is: -.79 b.778 c.898 d. e.89 [VCAA 6] e 6 Mths Quest Mthemticl Methods CAS for the Csio ClssPd
ETEnDED REsPonsE For the function f( )= 5 : i find the -intercept ii find the vlues f () nd f ( ) iii find the eqution of the smptote iv sketch the grph of f () v stte the domin nd rnge. b For the function g(), where g () = f ( + ) nd f () = 5 : i stte the trnsformtions to chnge f () to g () ii stte the eqution of the smptote iii sketch the grph of g () iv stte the domin nd rnge of g (). The number of lions, L, in wildlife prk is given b L = (.t ), where t is the number of ers since counting strted. At the sme time the number of cheeths, C, is given b C = 5(.5t ). Find the number of: i lions ii cheeths when counting begn. b Find the numbers of ech fter i er ii 8 months. c Which of the nimls is the first to rech popultion of nd b how long? d After how mn months re the popultions equl, nd wht is this popultion? The grph of the function f : R R, where f () = A +, is shown t right. Give the eqution of the horizontl smptote in the form = c. b The grph psses through the point (,.8). Use this informtion to find c the vlue of A. c Find the -intercept, correct to deciml plces. (,.8) d Find the vlues of: i f ( ) ii f (). e Find the vlue of if f () = 5.. f Stte the domin nd rnge of f (). The temperture T ºC of coffee in cermic mug t time t minutes fter it is poured is given b T = 6( 5. t ) +. Find the initil temperture of the coffee. b Find the temperture of the coffee, correct to one deciml plce, t: i minutes fter it is poured ii 5 minutes fter it is poured. c Sketch the grph of the eqution for t 5. If the coffee cn be comfortbl drunk when it is between tempertures of ºC nd 5 ºC, find: d the time vilble to drink the coffee e the finl temperture the coffee will settle to. 5 The number of bcteri (N) in culture is given b the eponentil function N = (.5t ), where t is the number of ds. Find the initil number of bcteri in the culture. b Find the number of bcteri in the culture fter: i ds ii weeks. c Find the time tken for the bcteri to rech. When the bcteri rech certin number, the re treted with n nti-bcteril serum. The serum destros bcteri ccording to the eponentil function D = N.789t, where D is the number of bcteri remining fter time t nd N is the number of bcteri present t the time the serum is dded. The culture is considered cured when the number of bcteri drops below. d If the bcteri re treted with the serum when their numbers rech, find the number of ds it tkes for the culture to be clssed s cured. e How much longer would it tke the culture to be cured if the serum is pplied fter 6 weeks? ebookplus Digitl doc Test Yourself Chpter 5 Chpter 5 Eponentil nd logrithmic functions 65
ebookplus CTIVITIEs Chpter opener Digitl doc Quick Questions: Wrm up with ten quick questions on eponentil nd logrithmic functions (pge 6) 5B Negtive nd rtionl powers Tutoril WE 5 int-9: Wtch tutoril on writing eponentil epressions with positive indices (pge ) Digitl doc SkillSHEET 5.: Prctise working with negtive nd rtionl powers (pge 5) 5C Indicil equtions Tutoril WE int-9: Wtch tutoril on solving n indicil eqution b using substitution (pge 7) 5D Grphs of eponentil functions Digitl docs SkillSHEET 5.: Prctise substituting vlues into eponentil functions (pge ) WorkSHEET 5.: Write epressions with their simplest inde nottion, solve indicil equtions nd sketch grphs of eponentils (pge ) 5F Solving logrithmic equtions Digitl doc WorkSHEET 5.: Simplif logrithmic epressions nd solve logrithmic equtions (pge 5) 5G Logrithmic grphs Interctivit Logrithmic grphs int-6: Consolidte our understnding of logrithmic grphs nd their fetures (pge 5) 5H Applictions of eponentil nd Digitl docs logrithmic functions Spredsheet : Investigte grphs of eponentil functions (pge 5) Etension: Prctise sketching logrithmic grphs nd identifing rules of functions (pge 5) Tutoril WE6 int-9: Wtch tutoril on clculting the price of gold, where the price is modelled b logrithmic function (pge 5) Chpter review Digitl doc Test Yourself: Tke the end-of-chpter test to test our progress (pge 65) To ccess ebookplus ctivities, log on to www.jcplus.com.u 66 Mths Quest Mthemticl Methods Cs for the Csio ClssPd