Unit- Exponential and Log Functions

7/24/2019 Unit- Exponential and Log Functions

1/48

Precalculus Name:____________________________________Prequiz- Logs and Exponential Functions

Date:_____________________________________

MULTIPLECHOICE(INDICATEYOURANSWERSINTHESPACEPROVIDED):

(1) !e expression (-"x#$")"is equi%alent to:(1)

(a) -&x'$& () -#x*$' (c) -#x'$& (d) -"x*$'

(#) +impli,$: )3()3( 4a2a +

(#)

(a) 4aa2

3 ++ ()4a33 + (c) 6a23 + (d) a4a

3

3 +

(") !at is axlog3 = .ritten in exponential ,orm/(")

(a) "x0 a () a"0 x (c) ax0 " (d) "a0 x

() !e equation $ 0 axexpressed in logarit!mic ,orm is:()

(a) ylogx a= () yloga x= (c) alogx y= (d) xlogy a=

(*) !e expression log 1# is equi%alent to:(*)

(a) log " 2 # log #() log ' 2 log '

(c) log " log (d) log " 3 # log #

(') !e expression log x is equi%alent to:(')

(a) log x() log 2 log x(c) (log )(log x)(d) log x


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() !e expressionb

alog

3

is equi%alent to:

()

(a)

blog

alog3

() )bloga(log3

1

(c) )bloga(log3 (d) blogalog3

(4) !e expression blog2

1alog + is equi%alent to:

(4)

(a) )balog( +

() balog

(c)( )

blog

2

1alog

(d) ablog

(&) 5, 6 0 pr #7 .!ic! equation is true/(&)

(a) rlog2pAlog +=

() )r(logp2Alog =(c) rlog2logplogAlog ++=

(d) rlog2plogAlog +=

(18) !ic! o, t!e ,ollo.ing equations is equi%alent to 5log33log73logx =+ /(18)

(a) 3x7 53 =() 125)7x( 3 =+

(c) 37x 53 =+ (d) 1521x3 =+

Precalculus Name:__________________________________ Lesson- properties7 equations .it! exponents and

po.er and exponential ,unctions Date:___________________________________

#


3/48

Objectives: use t!e properties o, exponents sol%e equations containing rational exponents examine po.er and exponential ,unctions

Do No: 9se t!e exponential properties to simpli,$ and re.rite t!e ,ollo.ing expressions:

(1) = yx aa

(2) ( ) =yxa

(3) ( ) =xab

(4) =

x

b

a

(5) =y

x

a

a

(6) =xa

(7) =0a

__________________________________________________________________________________________

5n +mall roups: 9se eac! example in t!e ;Do No.< to arri%e at general rules as t!e$ appl$ to monomials .it!exponents=

Usi!" E#$o!e!ti%& '!ctio! Po$eties to So&ve *o #:

Process 1 Process #

Examples (eac! relates to ;Process 1


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Po.er ,unction:

exponential ,unction:

+mall roup 6cti%it$

?n $our grap!ing calculator7 simultaneousl$ grap!: $ 0 8=*x7 $ 0 8=*x7 $ 0 #x7 $ 0 *x

(1) !at is t!e range o, eac! exponential ,unction/

(2) !at is t!e e!a%ior o, eac! grap!/

(3) Do t!e grap!s !a%e an$ as$mptotes/

(4) (a) !at point is on t!e grap! o, eac! ,unction/

() !$/

C+%%cteistics o* "%$+s o* , - !#

! . / 0 1 ! 1 /

domain

range

$-intercept

e!a%ior

!orizontal as$mptote

%ertical as$mptote

Extension: rap! t!e exponential ,unctions $ 0 #x7 $ 0 #x2 "7 and $ 0 #x3 # on t!e same set o, axes=@ompare and contrast t!e grap!s using a tale similar to t!e one ao%e=

Precalculus Name:__________________________________ Lesson- rap!ing exponential ,unctions7 exponential

gro.t! and deca$ Date:___________________________________


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Objectives: grap! exponential ,unctions use exponential ,unctions to determine gro.t! and deca$

Usi!" E#$o!e!ti%& '!ctio!s *o Re%& Wo&2 A$$&ic%tio!s:

Exponential gro.t!:

Exponential deca$:

Exponential ro.t! or Deca$: N 0 N8 (1 2 r)t

(1) rite a ,ormula t!at represents t!e a%erage gro.t! o, t!e population o, a cit$ .it! a rate o, =*A per $ear=Let x represent t!e numer o, $ears7 $ represent t!e most recent total population o, t!e cit$7 and 6 is t!ecit$Bs population no.= !at is t!e expected population in 18 $ears i, t!e cit$Bs population no. is ##7*8

people/ rap! t!e ,unction ,or 8 x #8=

*


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__________________________________________________________________________________________(2) +uppose t!e %alue o, a computer depreciates at a rate o, #*A a $ear= Determine t!e %alue o, a laptop

computer t.o $ears a,ter it !as een purc!ased ,or C"7*8=

(3) >exico !as a population o, aout 188 million people7 and it is estimated t!at t!e population .ill doule in#1 $ears= 5, population gro.t! continues at t!e same rate7 .!at .ill e t!e population in:(a) 1* $ears(b) "8 $ears

(c) grap! t!e population gro.t! ,or 8 time *8

__________________________________________________________________________________________(4) 6 researc!er estimates t!at t!e initial population o, !one$ees in a colon$ is *88= !e$ are increasing at a

rate o, 1A per .ee= !at is t!e expected population in ## .ees/

'


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(5) 5n 1&&87 Exponential @it$ !ad a population o, 887888 people= !e a%erage $earl$ rate o, gro.t! is *=&A=Find t!e proected population ,or #818=

(6) Find t!e proected population o, eac! location in #81*:

(a) 5n onolulu7 a.aii7 t!e population .as 4"'7#"1 in 1&&8= !e a%erage $earl$ rate o, gro.t! is8=A=

(b) !e population in Gings @ount$7 Ne. Hor !as demonstrated an a%erage decrease o, 8=*A o%erse%eral $ears= !e population in 1&& .as #7#87"4=


8/48

Precalculus Name:____________________________________Lesson- >ore exponential ,unction grap!s7

Population gro.t!7 !al,-li,e

Objectives: grap! exponential ,unctions use exponential ,unctions to determine population gro.t! and !al,-li,e deca$

(1) !e population o, Los 6ngeles @ount$ .as &71*7#1& in 1&&= 5, t!e a%erage gro.t! rate is 8=*A7 predictt!e population in #818=

rap! t!e equation ,or 8 time #8=

(2) Iadioacti%e gold 1&4 (1&46u)7 used in imaging t!e structure o, t!e li%er7 !as a !al,-li,e o, #=' da$s= 5, t!einitial amount is *8 milligrams o, t!e isotope7 !o. man$ milligrams (rounded to the nearest tenth).ill ele,t o%er a,ter:

(a) J da$(b) 1 .ee

4


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(3) 5, a ,armer uses #* pounds o, insecticide7 assuming its !al,-li,e is 1# $ears7 !o. man$ pounds (rounded tothe nearest tenth).ill still e acti%e a,ter:

(a) * $ears(b) #8 $ears

(4) 5n #8887 t!e c!icen population on a ,arm .as 187888= !e numer o, c!icens increased at a rate o, &Aper $ear= Predict t!e population in #88*=

rap! t!e equation ,or 8 time 1*=

(5) 5, Gen$a !as a population o, aout "878887888 people and a douling time o, 1& $ears and i, t!e gro.t!continues at t!e same rate7 ,ind t!e population (rounded to the nearest million) in:

(a) 18 $ears(b) "8 $ears

&


10/48

Precalculus Name:__________________________________ Lesson- @ompound 5nterest

Date:___________________________________

Objectives: use exponential functions to determine compound interest

Do No:

(1) 6 laser printer .as purc!ased ,or C"88 in #881= 5, its %alue depreciates at a rate o, "8A a $ear7 determine!o. muc! it .ill e .ort! in #88=

(2) Iates can e compounded in di,,erent increments per $ear= Exponential gro.t! occurs !o. o,ten i, t!e rate

is compounded:

annuall$:

i-annuall$:

quarterl$:

mont!l$:

.eel$:

dail$:

!e general equation ,or exponential gro.t! is modi,ied ,or ,inding t!e alance in an account t!at earnscompound interest=

@ompound 5nterest:tn

n

r1PA

+=

18


11/48

__________________________________________________________________________________________(1) 5, @!arlie in%ested C17888 in an account pa$ing 18A compounded mont!l$7 !o. muc! .ill e in t!e

account at t!e end o, 18 $ears/

(2) >ie .ould lie to !a%e C#87888 cas! ,or a ne. car * $ears ,rom no.= o. muc! s!ould e placed in anaccount no. i, t!e account pa$s &=*A compounded .eel$/

(3) +uppose C#7*88 is in%ested at A compounded quarterl$= o. muc! mone$ .ill e in t!e account in:(c) K $ear(d) 1* $ears

__________________________________________________________________________________________(4) +uppose C7888 is in%ested at 11A compounded .eel$= o. muc! mone$ .ill e in t!e account in:(e) J $ear(f) 18 $ears

11


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(5) o. muc! mone$ must @ind$ in%est ,or a ne. $ac!t i, s!e .ants to !a%e C*87888 in !er account t!at earns*A compounded quarterl$ a,ter 1* $ears/

(6) @arol .on C*7888 in a ra,,le= +!e .ould lie to in%est !er .innings in a mone$ maret account t!atpro%ides an 6PI o, 'A compounded quarterl$= Does s!e !a%e to in%est all o, it in order to !a%e C&7888 int!e account at t!e end o, 18 $ears/ +!o. $our .or and explain$our ans.er=

1#


13/48

Precalculus Name:__________________________________ Lesson: Exponential Functions .it! ase e

Date:___________________________________

Objective: use exponential ,unctions .it! ase e

Euler +a%ings an pro%ides a sa%ings account t!at earns compounded interest at a rate o, 188A= Hou ma$

c!oose !o. o,ten to compound t!e interest7 ut $ou can onl$ in%est C1 o%er t!e course o, one $ear=

1"


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Exponential ro.t! or Deca$ (in terms o, e): N 0 N8 et

(1) 6ccording to Ne.ton7 a eaer o, liquid cools exponentiall$ .!en remo%ed ,rom a source o, !eat=

6ssume t!at t!e initial temperature iis &8F and t!at 0 8=#*=

(a) rite a ,unction to model t!e rate at .!ic! t!e liquid cools=

() Find t!e temperature o, t!e liquid a,ter minutes (t)

(c) rap! t!e ,unction and use t!e grap! to %eri,$ $our ans.er in part ()

1


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(2) +uppose a certain t$pe o, acteria reproduces according to t!e model 0 188 e8=#1 t 7 .!ere t is t!etime in !ours=

(a) 6t .!at percentage rate does t!is t$pe o, acteria reproduce/

() !at .as t!e initial numer o, acteria/

(c) Find t!e numer o, acteria (rounded to the nearest whole number)a,ter:(i) * !ours(ii) 1 da$(iii) " da$s

(3) 6 cit$Bs population can e modeled $ t!e equation $ 0 ""7"8e8=8"& t 7 .!ere t is t!e numer o, $ears since1&*8=

(a) as t!e cit$ experienced a gro.t! or decline in population/

() !at .as t!e population in 1&*8/

(c) Find t!e proected population in #818=

1*


16/48

Precalculus Name:__________________________________ - @ompound 5nterest

Date:___________________________________

(1) 5, $ou in%est C*7#*8 in an account pa$ing 11="4A compounded continuousl$7 !o. muc! mone$ .ill e int!e account at t!e end o,:

(a) ' $ears " mont!s(b) #8 mont!s

(2) 5, $ou in%est C7*88 in an account pa$ing 4="*A compounded continuousl$7 !o. muc! mone$ .ill e int!e account at t!e end o,:

(a) *=* $ears(b) 1# $ears

1'


17/48

__________________________________________________________________________________________(3) 6 promissor$ note .ill pa$ C"87888 at maturit$ 18 $ears ,rom no.= o. muc! s!ould $ou e .illing to

pa$ ,or t!e note no. i, t!e note gains %alue at a rate o, &A compounded continuousl$/

(4) +uppose Nii deposits C17*88 in a sa%ings account t!at earns '=*A interest compounded continuousl$=+!e plans to .it!dra. t!e mone$ in ' $ears to mae a C#7*88 do.n pa$ment on a car= ill t!ere eenoug! ,unds in NiiBs account in ' $ears to meet !er goal/ Explain $our ans.er=

1


18/48

Precalculus Name:__________________________________ Lesson- @ontinuous @ompound 5nterest

Date:___________________________________

Objective: use exponential ,unctions to determine continuousl$ compounded interest

@ontinuousl$ @ompounded 5nterest: 6 0 Pert

(1) im and Gerr$ are sa%ing ,or t!eir daug!terBs college education= 5, t!e$ deposit C1#7888 in an accountearing '=A interest compounded continuousl$7 !o. muc! .ill e in t!e account .!en s!e goes to collegein 1# $ears/

(2) Paul in%ested a sum o, mone$ in a certi,icate o, deposit t!at earns 4A interest compounded continuousl$=5, Paul made t!e in%estment on Manuar$ 17 1&&*7 and t!e account .as .ort! C1#7888 on Manuar$ 17 1&&&7.!at .as t!e original amount in t!e account/

14


19/48

(3) @ompare t!e alance a,ter "8 $ears o, a C1*7888 in%estment earning 1#A interest compoundedcontinuousl$ to t!e same in%estment compounded quarterl$=

(4) i%en t!e original principal7 t!e annual interest rate7 t!e amount o, time ,or eac! in%estment7 and t!e t$peo, compounded interest7 ,ind t!e amount at t!e end o, t!e in%estment:

(a) P 0 C17#*8 r 0 4=*A t 0 " $ears compounded semi-annuall$

(b) P 0 C#7** r 0 '=#*A t 0 * $ears " mont!s compounded continuousl$

1&


20/48

Precalculus Name:__________________________________ - @ompound 5nterest

Date:___________________________________

(1) 5, $ou in%est C*7#*8 in an account pa$ing 11="4A compounded continuousl$7 !o. muc! mone$ .ill e int!e account at t!e end o,:

(a) ' $ears " mont!s(b) #8 mont!s

(2) 5, $ou in%est C7*88 in an account pa$ing 4="*A compounded continuousl$7 !o. muc! mone$ .ill e int!e account at t!e end o,:

(a) *=* $ears(b) 1# $ears

#8


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__________________________________________________________________________________________(3) 6 promissor$ note .ill pa$ C"87888 at maturit$ 18 $ears ,rom no.= o. muc! s!ould $ou e .illing to

pa$ ,or t!e note no. i, t!e note gains %alue at a rate o, &A compounded continuousl$/

(4) +uppose Nii deposits C17*88 in a sa%ings account t!at earns '=*A interest compounded continuousl$=+!e plans to .it!dra. t!e mone$ in ' $ears to mae a C#7*88 do.n pa$ment on a car= ill t!ere eenoug! ,unds in NiiBs account in ' $ears to meet !er goal/ Explain $our ans.er=

#1


22/48

Precalculus Name:____________________________________Lesson- Properties o, a logs7 re.riting

Exponential ,unctions as logarit!ms7 log grap!s Date:_____________________________________

Objective:

o learn .!at a logarit!m is o learn t!e properties o, logs o learn to re.rite an exponential ,unction as a logarit!m rap!ing logs

Do No.: +ol%e ,or x: 1&" += xx and c!ec=

_________________________________________________________________________________________W+%t is % &o"%it+34

Logarit!ms are in%erses o, exponential ,unctions= Logarit!ms are ,unctions ecause exponential ,unctions areone-to-one ,unctions=

e cannot sol%e an equation lie: xy #= using t!e algeraic tec!niques .e !a%e learned so ,ar= !ere,ore7 .emust tr$ an alternati%e tec!nique=

R&e: xytoequivalentisbx by log==

!e log to t!e ase is t+e e#$o!e!tto .!ic! must e raised to otain x=

Po$eties o* Lo"s

01logb =

1blogb =

xblog x

b =

xb xlog b

= 7 .!ere x O 8

NlogMlogMNlog bbb +=

NlogMlogN

Mlog bbb =

MlogpMlog bp

b =

Example:

@on%ert eac! into logarit!mic ,orm @on%ert eac! into logarit!mic ,orm

1= xy #= =#

1*log #* =

#= &" = *= cba =log

"=1*

*

1 = '= #&

1log" =

##


23/48

__________________________________________________________________________________________W+%t is % N%t%& Lo"%it+34

R&e: xytoequivalentisbx by log==

!e log to t!e ase is t+e e#$o!e!tto .!ic! must e raised to otain x=

Po$eties o* Lo"s81ln =

1ln =b

xex =ln

xe x =ln 7 .!ere x O 8NMMN lnlnln +=

NMN

Mlnlnln =

MpM p lnln =

Example:

@on%ert eac! into logarit!mic ,orm @on%ert eac! into logarit!mic ,orm

1= xey= = x=*ln#= xe= *= cb=ln

"=11 = e

e'= #ln =y

Example:

rap! eac! o, t!e ,ollo.ing on t!e same set o, axes using t!e grap!ing calculator=

yx

xy

x

yy

x

==

=

=

#

#

log=

log="

#=#

#=1

yx

xy

ex

ey

y

x

=

=

=

=

ln=4

ln=)

='

=*

Precalculus Name:__________________________________ Lesson- +impli,$ log expressions7 common logs7 e%aluate

Date:___________________________________

Objectives: simpli,$ expressions using t!e properties o, logarit!mic ,unctions de,ine common logarit!ms

#"

x

$


24/48

e%aluate expressions in%ol%ing logarit!ms

Pob&e3 Set:.rite t!e ,ollo.ing expressions in simpler logarit!mic ,orms:

(1) 72

b vlog (2) 2b a

1log

(3)

2

1

3

2

b

n

!log (4)

v"

logb

(5) xlogb (6) 32

3

b#p

nlog

(7) 9se logarit!mic properties to ,ind t!e %alue o, x (.it!out using a calculator):

6log$log3

2%log

2

1xlog

bbbb

+=

#


25/48

rite eac! expression in terms o, a single logarit!m .it! a coe,,icient o, one:v

logvloglog2&'e

2

bbb =

($) ylog4xlog5 bb + (%) ylogxlog2 bb

(10) log4

1ylog2xlog3 bbb + (11) clog$ b

(12) log2"log2

3bb (13) )ba(log

3

1 32b +

@ommon Logarit!m:

xloglog18=

@!ange o, ase Formula:

b

a

b

a

b

aa

p

p

blog

log

ln

ln

log

loglog ===

i%enlogan7 e%aluate eac! logarit!m to ,our decimal places:

(14) 172log$ (15) 25$1log6 (16) 00650log13

Extension: i%en $ 0 logn7 .!at can $ou determine aout t!e log %alue ($) ased on and n/

#*


26/48

Precalculus Name:____________________________________Lesson- Properties o, Logarit!mic Functions7

+impli,$ing logarit!mic expressions Date:_____________________________________

Objective: examine properties o, logarit!mic ,unctions simpli,$ expressions using t!e properties o, logarit!mic ,unctions

9se t!e properties o, logarit!mic ,unctions to sol%e ,or x:

(1) 2xlog5 = (2) x64log4 =

(3) 3$logx = (4)3

2xlog$ =

9se t!e properties o, logarit!mic ,unctions to simpli,$ eac! expression:

(5) $log$ (6) 1log 5)0

(7) 000*1log10 ($) 64log2

(%) 343log7 (10) 0010log10

(11) eloge (12) 3

5 5log

#'


27/48

rite t!e ,ollo.ing expressions in simpler logarit!mic ,orms:

(13) %6

b yxlog (14)$

7

b

vlog

(15)p#

!nlogb (16) 4b a

1log

(17) 5b xlog (1$) 3 22

b yxlog

Precalculus Name:____________________________________

#


28/48

6cti%it$- rap!ing Log EquationsQQill e collected and graded (separate paper) Date:_____________________________________

Objective: o learn !o. to grap! log equations t!at are not o, ase 18 or e=

D? N?: Find log" to t!e nearest ten-t!ousandt!s place=

__________________________________________________________________________________________

5= rap! eac! o, t!e ,ollo.ing on t!e same set o, coordinate axes and ans.er t!e ,ollo.ing questions=

1= )1(log# += xy#= )1(log" += xy

"= )1(log += xy= )1(log* += xy

a= !at are some notale similarities and di,,erences among t!e grap!s/

= !at appears to !appen as t!e ase gets larger and larger/

55= rap! eac! o, t!e ,ollo.ing on t!e same set o, coordinate axes and ans.er t!e ,ollo.ing questions=

1= )1(log

# += xy

#= )#(log# += xy"= )"(log# += xy= )(log# += xy

a= !at are some notale similarities and di,,erences among t!e grap!s/

= !at appears to !appen as t!e constant in t!e inomial c!anges/

p"1" R-4* odd7 &-188 all

Precalculus Name:__________________________________

#4


29/48

Lesson- Natural Log ord ProlemsDate:___________________________________

Objectives: sol%e real-.orld applications .it! natural logarit!mic ,unctions

Do No:

Laura .on C#7*88 on a game s!o.= +!e .ould lie to in%est !er .innings in an account t!at earns an interestrate o, 1#A compounded continuousl$= Does s!e !a%e to in%est all o, it in order to !a%e C7888 in t!e account

at t!e end o, $ears to put a do.n pa$ment on a ne. sailoat/ +!o. $our .or and explain$our ans.er=

(1) 6na is tr$ing to sa%e ,or a ne. !ouse= o. man$ $ears7 to t!e nearest $ear7 .ill it tae 6na to triple t!emone$ in !er account i, it is in%ested at A compounded annuall$/

(2) 6t .!at annual percentage rate (to t!e nearest !undredt! o, a percent) compounded continuousl$ .illC'7888 !a%e to e in%ested to amount to C117888 in 4 $ears=

#&


30/48

__________________________________________________________________________________________(3) 5n 1&&87 Exponential @it$ !ad a population o, 1#7888 people= 5n .!at $ear .ill t!e cit$ !a%e a

population o, aout #887888 people i, it .as gro.ing at an exponential rate o, 0 8=81/

(4) 5, C*7888 is in%ested at an annual interest rate o, *A compounded quarterl$7 !o. long .ill it tae t!ein%estment to doule/

(5) !at .as t!e annual interest rate (to t!e nearest !undredt! o, a percent) o, an account t!at too 1# $earsto doule i, t!e interest .as compounded continuousl$ and no deposits or .it!dra.als .ere made duringt!e 1#-$ear period/

"8


31/48

Precalculus Name:__________________________________ Lesson- >ore natural log .ord prolems

Date:___________________________________

Objective: sol%e real-.orld applications .it! natural logarit!mic ,unctions

(1) 5, a car originall$ costs C147888 and t!e a%erage rate o, depreciation is "8A7 ,ind t!e %alue o, t!e car to t!enearest dollar a,ter ' $ears=

(2) o. man$ $ears7 to t!e nearest $ear7 .ill it tae ,or t!e alance o, an account to doule i, it is gaining 'A

interest compounded semiannuall$/

(3) !en Iac!el .as orn7 !er mot!er in%ested C*7888 in an account t!at compounded A interest mont!l$=Determine t!e %alue o, t!is in%estment .!en Iac!el is #* $ears old=

"1


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__________________________________________________________________________________________

(4) !e deca$ o, caron-1 can e descried $ t!e ,ormulat0001240

0eAA = = 9sing t!is ,ormula7 !o. man$

$ears7 to t!e nearest $ear7 .ill it tae ,or caron-1 to diminis! to 1A o, t!e original amount/

(5) 5n #88#7 a ,armer !ad 88 pigs on !is ,arm= e estimated t!at t!is population o, pigs .ill doule in 1*$ears= 5, population gro.t! continues at t!e same rate7 predict t!e numer o, pigs in:

a= #818= #8"8

(6) 5, t!e .orld population is aout ' illion people no. and i, t!e population gro.s continuousl$ at an annualrate o, 1=A7 .!at .ill t!e population e (to t!e nearest illion) in 18 $ears ,rom no./

"#


33/48

__________________________________________________________________________________________(7) 5, C188 is in%ested in an account t!at !as an interest o, A compounded quarterl$7 !o. long .ill it tae ,or

t!e alance to reac! a %alue o, C17888/

($) !at interest rate (to t!e nearest !undredt! o, a percent) compounded mont!l$ is required ,or an C47*88in%estment to triple in * $ears/

(%) 6n optical instrument is required to oser%e stars e$ond t!e sixt! magnitude7 t!e limit o, ordinar$ %ision=o.e%er7 e%en optical instruments !a%e t!eir limitations= !e limiting magnitude L o, an$ opticaltelescope .it! lens diameter D7 in inc!es7 is gi%en $ t!e equation +log15$$, += = 9se t!is equationto ,ind t!e ,ollo.ing to t!e nearest tenth:

a= t!e limiting magnitude ,or a !omemade '-inc! re,lecting telescope== t!e diameter o, a lens t!at .ould !a%e a limiting magnitude o, #8='=

""


34/48

9nit ': Exponential S Logarit!mic Functions

Definitions, Properties & Formulas

Po$eties o* E#$o!e!ts

Po$et, De*i!itio!

Product baba xxx +=

Tuotient bab

a

xx

x = 7 .!ere x 8

Po.er Iaised to a Po.er (xa)0 xa

Product Iaised to a Po.er (x$)a0 xa $a

Tuotient Iaised to a Po.er a

aa

y

x

y

x=

7 .!ere $ 8

Uero Po.er x80 17 .!ere x 8

Negati%e Po.ern

n

x

1x = 7 .!ere x 8

Iational Exponent

nn

1

xx =

,or an$ real numer x 8 and an$ integer n O 1and .!en x V 8 and n is odd

E#$o!e!ti%&

5ot+6Dec%,

N 0 N8 (1 2 r)t

.!ere: N is t!e ,inal amount7 N8is t!e initial amount7 t is t!e numer o, timeperiods7 and r is t!e averagerate o, gro.t!(positi%e) or deca$(negati%e) pertime period

Co3$o!2

I!teest (Peio2ic)

tn

n

r1PA

+=

.!ere: 6 is t!e ,inal amount7 P is t!e principal in%estment7 r is t!e annualinterest rate7 n is t!e numer o, times interest is compounded eac! $ear7 and t ist!e numer o, $ears

E#$o!e!ti%&

5ot+6Dec%,

(i! te3s o* e)

N 0 N8 et

.!ere: N is t!e ,inal amount7 N8is t!e initial amount7 t is t!e numer o, timeperiods7 and (a constant) is t!e exponentialrate o, gro.t!(positi%e) or

deca$(negati%e) per time period

Co!ti!os&,

Co3$o!2e2

I!teest

6 0 Pert

.!ere: 6 is t!e ,inal amount7 P is t!e principal in%estment7 r is t!e annualinterest rate7 and t is t!e numer o, $ears

"


35/48

Lo"%it+3ic

'!ctio!s

are in%erses o, exponential ,unctions

a logarit!m is an exponentW

Co33o!

Lo"%it+3s

.!en no ase is indicated7 t!e ase is assumed to e 18

xlogxlog 10=

x10yxlog y ==

C+%!"e o* 7%se

'o3&%alog

nlognlog

b

ba =

.!ere a7 7 and n are positi%e numers7 and a 17 1

N%t%&

Lo"%it+3s

instead o, log7 ln is used t!ese logarit!ms !a%e a ase o, e

lnx xloge=

lnx 0 $ xey =

all properties o, logarit!ms also !old ,or natural logarit!ms

Po$eties o* Lo"%it+3ic '!ctio!s

5, 7 >7 and N are positi%e real numers7 17 and p and x are real numers7 t!en:

De*i!itio! E#%3$&es

01logb = written exponentially: 80 1

1blogb = written exponentially: 10

xblog x

b = written exponentially: x0 x

xb xlog b

= 7 .!ere x O 8 710 7log 10

=

NlogMlogMNlog bbb +=xlog%logx%log 333 +=

(logylogy(log5

1

5

1

5

1+=

NlogMlogN

Mlog bbb =

5log2log5

2log 444 =

xlog7logx

7log $$$ =

MlogpMlog bp

b =

6logx6log 2x

2 =

ylog4ylog 54

5 =

NlogMlog bb = i, and onl$ i, > 0 N)2x5(log)4x3(log 66 +=

)2x5()4x3( +=

"*


36/48

Po$eties o* Lo"%it+3ic '!ctio!s

5, 7 >7 and N are positi%e real numers7 17 and p and x are real numers7 t!en:

De*i!itio! E#%3$&es

01logb = written exponentially: 80 1

1blogb = written exponentially: 10

xblog

x

b = written exponentially: x

0 x

xb xlog b

= 7 .!ere x O 8 710 7log 10

=

NlogMlogMNlog bbb +=xlog%logx%log 333 +=

(logylogy(log5

1

5

1

5

1+=

NlogMlogN

Mlog bbb =

5log2log5

2log 444 =

xlog7logx

7log $$$ =

MlogpMlog bp

b = 6logx6log2

x

2 =

ylog4ylog 54

5 =

NlogMlog bb = i, and onl$ i, > 0 N)2x5(log)4x3(log 66 +=

)2x5()4x3( +=

Co33o! Eos:

NlogMlogNlog

Mlogbb

b

b N

MlogNlogMlog bbb =

Nlog

Mlog

b

b

cannot e simpli,ied

NlogMlog)NM(log bbb ++ MNlogNlogMlog bbb =+

)NM(logb + cannot e simpli,ied

Mlogp)M(log bp

b p

bb MlogMlogp =p

b )M(log cannot e simpli,ied

Precalculus Name:____________________________________Ie%ie.- Exponential and Logarit!mic Functions part 1

Date:_____________________________________

"'


37/48

ANSWERTHE'OLLOWIN58UESTIONSONASEPARATESHEETO'PAPERANDSHOWALLWOR9

rite eac! expression in terms o, simpler logarit!mic ,orms:

(1) yxlog 5

b

(2)7

5

4

b

-log (3) $b c

1log (4)

p

n!log

35

b

i%enlogan7 e%aluate eac! logarit!m to ,our decimal places:

(5) 42log3 (6) 5log

2

1 (7) 000%$0log6

+ol%e eac! equation and round ans.ers to ,our decimal places .!ere necessar$:

($) 3xlog2 = (%) 36logxlog4log 555 =+

(10) x50e751000= (11) 2xlog 6 =

(12) x4%

1log7 = (13)

2

14logx =

(14) 52710x = (15) )3xlog(2log5logxlog =

(16) 12logxlog =+ (17) 3xlog4 =

(1$) 2log3)x5(log %% = (1%) 1xlog20log =

(20) x400212= (21) 251e x25 =

(22) 2)5xlog()10xlog( =++(23) xlog36log

2

1216log 666 =

Precalculus Name:____________________________________Ie%ie.- Exponential and Logarit!mic Functions part #

Date:_____________________________________SHOWALLWOR9:

"


38/48

(1) 6nt!on$ is an actuar$ .oring ,or a corporate pension ,und= e needs to !a%e C1=' million gro. to C##million in ' $ears= !at interest rate (to t!e nearest !undredt! o, a percent) compounded annuall$ does !eneed ,or t!is in%estment/

(2) !e numer o, guppies li%ing in Logarit!m Lae doules e%er$ da$= 5, t!ere are ,our guppies initiall$:c= Express t!e numer o, guppies as a ,unction o, t!e time t=d= 9se $our ans.er ,rom part (a) to ,ind !o. man$ guppies are present a,ter 1 .ee/

e= 9se $our ans.er ,rom part (a) to ,ind7 to t!e nearest da$7 .!en .ill t!ere e #7888 guppies/

"4


39/48

SHOWALLWOR9:

(3) !e relations!ip et.een intensit$7 i7 o, lig!t (in lumens) at a dept! o, x ,eet in Lae Erie is gi%en $

x00235012

'log = = !at is t!e intensit$7 to t!e nearest tent!7 at a dept! o, 8 ,eet/

(4) ii .ent to a roc concert .!ere t!e deciel le%el .as 44= !e deciel is de,ined $ t!e ,ormula

0'

'log10+= 7 .!ere D is t!e deciel le%el o, sound7 i is t!e intensit$ o, t!e sound7 and i80 18 -1#.att per

square meter is a standardized sound le%el= 9se t!is in,ormation and ,ormula to ,ind t!e intensit$ o, t!esound at t!e concert=

"&


40/48

SHOWALLWOR9:

(5) o. man$ $ears7 to t!e nearest $ear7 .ill it tae t!e .orld population to doule i, it gro.s continuousl$ atan annual rate o, #A=

(6) an 6 pa$s 4=*A interest compounded annuall$ and an pa$s 4A interest compounded quarterl$= 5,$ou in%est C*88 o%er a period o, * $ears7 .!at is t!e di,,erence in t!e amounts o, interest paid $ t!e t.oans/

(7) Determine !o. muc! time7 to t!e nearest $ear7 is required ,or an in%estment to doule in %alue i, interest isearned at t!e rate o, *=*A compounded quarterl$=

8


41/48

Precalculus Name:__________________________ 6cti%it$- Ie%ie. o, Expoenentials and Logs

Date:___________________________

!e accompan$ing diagrams contain exponential and logarit!mic expressions and equations= !en cut out7 t!e14 equilateral triangles ,it toget!er to ,orm a large r!omus= For t!e triangles to create t!is s!ape7 t.oexpressions t!at are equi%alent must e touc!ing eac! ot!er7 s!aring t!e same edge= 6ll triangles must e usedto complete t!e r!omus= !ere are expressions t!at !a%e eit!er t!e same or similar ans.ers7 so c!ec $our.or and eac! pairing care,ull$ ot!er.ise $ou ma$ ,ind triangles t!at do not ,it properl$=

SHOW ALL WOR9 ON A SEPARATE SHEET O' PAPER

1


42/48

Precalculus Name:____________________________________est x # - inter Proect

Date:_____________________________________

Objective: o use exponential S log ,unctions to design a plan to sa%e C1 million as quicl$ as possile=

Rese%c+

Hou .ill need: XMo title7 description7 and salar$X+a%ings (assume interest rate is constant)5n%estment 5n,ormation

XLi%ing expenses (including utilities7 p!one7 groceries7 entertainment7 etc)XPlace to li%e (and amount o, rent and renterBs insurance or mortgage and taxesX@artransportation expensesX>iscellaneous expensesXPrior det (student loans7 etc)

M%t+

Hou .ill need to include: Xritten explanation o, $our scenario (t$ped7 doule spaced7 1# point NI ,ont)Xexponential and logarit!mic equations and t!eir solutions or Y> +ol%er DataXrap!s t!at model t!e rate o, pro,itincome gro.t!Xritten conclusion discussing t!e %iailit$ o, $our scenario

De

Frida$ Manuar$ &7 #88&

Hou .ill !a%e (#) class sessions e,ore t!e due date during .!ic! $ou ma$ conduct researc!7 as questions o,me7 conduct mat!ematical computations7 andor .or on t!e %eral portion o, t!e proect=

e .ill also !a%e (#) class sessions in a computer la .!ere .e .ill:1= Learn !o. to create >+ ord documents consisting o, mat!ematical equations#= e ale to conduct researc! ,or our proects=

5DE6+/

+tudent Loans-

+a%ings 6ccounts-

ransportation-

?.nIent ouse-

5nsurance-

Mos-

>iscellaneous-

#


43/48

Precalculus Name:____________________________________>odel- inter Proect @alculations (land)

Date:_____________________________________

Mo title7 description7 and salar$: >at! eac!er7 eac! >at!7 C*8 per mont! (used >edian career %alue)+a%ings: A +a%ings account7 deposit income 3 expense eac! mont!

Expenses t!at donBt go a.a$Electric7 as7 ?il: C88 per mont!

P!one: C* per mont!roceries: C*88 per mont!Entertainment: C"*8 per mont!Ient (including insurance): C1*88 per mont!@ar expenses (maintenance): C*8 per mont!@ar insurance: C188 per mont!

Expense t!at expires a,ter * $ears@ar Pa$ment: C""" per mont!

Expense t!at expires a,ter #8 $ears

+tudent loans: C"" per mont!

Prior +a%ingsC*87888

5 !ad t.o expenses t!at did not carr$ on ,ore%er= !ere,ore 5 decided to rea m$ proect up into p!ases=

P!ase 5 P!ase 55 P!ase 555*8 t Hears #8* < t Hears #8>t Hears

5ncome Expenditures

C*8 C88

C*C*88C"*8C1*88C*8C188C"""C""

+urplus o, C18'&mont! +urplus o, C18#mont! +urplus o, C1*mont!

P!ase 5: 6,ter * $ears7 5 no. !a%e a total o, C1"17"=" sa%edP!ase 55: 6,ter #8 $ears7 5 no. !a%e a total o, C*4*71*&="1#" sa%ed

o. long .ill it tae me to arri%e at a sa%ings o, C178887888/5 sol%ed ,or N in Y> +?LYEI and arri%ed at approximatel$ &=4*'441 mont!s e$ond t!e #8 t!$ear= !isgi%es me a total o, approximatel$ # $ears7 18 mont!s7 #* da$s7 1' !ours7 * minutes and 1' seconds to arri%eat C178887888 ased on t!is in,ormation=

5ncome Expenditures

C*8 C88C*C*88C"*8C1*88C*8C188

5ncome ExpendituresC*8 C88

C*C*88C"*8C1*88C*8C188C""

"


44/48

QQNote- .o %er$ important t!ings to e a.are o,: a= 5 ne%er got a raiseW Do $ou t!in $ou mig!t/ o.muc!/ !en/ %!2= !e costs in m$ scenario ne%er increasedW !at aout in,lation/ ig!er taxes7 etc/


45/48

Precalculus Name:____________________________________Lesson- >at! on >+?ID

Date:_____________________________________

Objective: o learn to use >icroso,t ord to create mat! related documents=

Example:

1= #)#(# # += xy

#=

?n Hour ?.n:

11'

)#(

#*

)#( ##

=

+ xy

"

"

" #

* *C

1*x

x

Exit icet- >+?IDQQPrint out and !and in at t!e end o, class

1=# #1# "' "'

' '

x x x

x

+

#=

8tan *& 01*

c

i%en: 5sosceles triangle @67

CT T 7!Tisects V @67

!Cand ! are dra.n

Pro%e: V+@6 +6@

*


46/48

+tatistics Name:____________________________________Lesson- Y> +ol%er

Date:_____________________________________

Objective: o learn !o. to use t!e Y> +ol%er on t!e 5-4"4 to determine exponential gro.t! and deca$as t!e$ appl$ to:

+a%ings accounts >ortgages +tudent loan repa$ment

__________________________________________________________________________________________

@ompound 5nterest Formula:

>ini Example:

_________________________________________________________________________________________

Fields in t!e Y>

!e %ariales listed in t!e Y> sol%er are called Z,ieldsZ= Eac! %ariale represents a quantit$ associated .it! acommon ,inanial concept or ,ormula=

N:!is represents t!e numer o, compounding periods in t!e term o, t!e in%estment7 annuit$ or loan=!is .ill al.a$s e a positi%e %alue=

I;:!is represents t!e Znominal rateZ ,or an in%estment7 annuit$ or loan= !is .ill al.a$s e a positi%e%alue= Note: e .rite t!e percent ,orm !ere7 not t!e ,raction or decimal ,orm o, a percent=

PV:!is represents t!e Zpresent %alueZ o, an in%estment7 loan or annuit$= !is numer can e positi%e ornegati%e= 5, t!e numer is positi%e7 t!en it indicated mone$ .as collected as in a loan= 5, t!e numer is

negati%e7 t!en it represents mone$ .e paid out7 as in an in%estment or loan .!ere .e are t!e lender= PMT:!is represents t!e pa$ment made to uild an annuit$ or pa$ o,, a loan= !e %alue .ill al.a$s e

negati%e in t!ese situations= 5, .e !a%e a Zpa$outZ annuit$7 t!en t!e %alue .ill e positi%e= 5n eit!er case7t!e %alue represents t!e pa$ment per compounding period=

'V:!is represents t!e Z,uture %alueZ o, an in%estment7 annuit$ or loan a,ter N compounding periods!a%e passed= !is %alue .ill e positi%e or negati%e depending on t!e signs o, PY and P>=

P6Y:!is %alue represents t!e numer o, pa$ments per $ear ,or annuities and loans= C6Y:!is represents t!e numer o, compounding periods per $ear= !ese must ot! e positi%e integers

greater t!an 1=

'


47/48

PMT: END 7E5IN +ol%er ,or Zordinar$Z annuities7 (END)7 orannuities ZdueZ (E5N)=

E# /: +ue +immons .ants to re-,inance !er !ouse= +!e currentl$ o.es C1#87888 and closing costs .ill eC7*88= +!e gets a "8-$ear mortgage at 'A nominal interest= o. large .ill !er mont!l$ pa$ment e/

Ex #: >ie maes an initial deposit in a ne. sa%ings account o, C187888= 5, t!is account accrues interest at arate o, "=&A compounded mont!l$ and >ie deposits C*88 per mont!7 !o. man$ $ears (to t!e nearestmont!) .ill it tae !im to !a%e C178887888 in !is account/

Ex ": Pro,essor [ !ad C*87888 in outstanding student loans at a '=*A interest rate upon ,inis!ing grad sc!ool=5, !e plans on pa$ing t!e loan o,, in 18 $ears7 .!at .ill !is mont!l$ pa$ment e/ o. muc! on totalinterest .ill !e !a%e paid at t!e end o, t!e 18 $ears/

O! Yo O!

1= Due plans on purc!asing a "I !ouse in +carsdale ,or C887888= e taes out a mortgage ,orC*87888 to pa$ ,or realtor expenses7 t!e ,irst ,e. mont!s o, utilities and taxes7 and ,or some minorcosmetic .or on t!e !ouse= !e mortgage !e quali,ies ,or is a "8 $ear loan at &A nominal interest=

!at .ill !is mont!l$ pa$ment e i, !e some!o. got a.a$ .it! putting C8 as a do.n pa$ment/ !at.ill !is mont!l$ pa$ment e i, !e put C1887888 as a do.n pa$ment/ o. muc! in total interest .ill epaid o%er t!e li,e o, t!e "8 $ear mortgage in eac! case/

N=

I%=

PV=

PMT=

FV=

P/Y=

C/Y=

PMT: END BEGIN

N=

I%=

PV=

PMT=FV=

P/Y=

C/Y=

PMT: END BEGIN

N=

I%=

PV=PMT=

FV=

P/Y=

C/Y=

PMT: END BEGIN

N=

I%=

PV=

PMT=

FV=

P/Y=

C/Y=

PMT: END BEGIN

N=

I%=

PV=

PMT=

FV=

P/Y=

C/Y=

PMT: END BEGIN


48/48

#= Ie%isit example " ,rom t!e lesson= Pro,essor [ decides to consolidate !is loans o%er a #8 $ear period at'A interest= o. muc! more in interest .ill !e !a%e paid t!an on t!e 18 $ear plan/

Extra @redit- * est Points (all or not!ing)+uppose $ou currentl$ li%e .it! $our parents ut .ould lie to purc!ase a !ouse o, $our o.n= Hou add

up all $our current mont!l$ expenses and sutract t!em ,rom $our mont!l$ net salar$ and disco%er aC17*88 surplus= Hou also !a%e C87888 in a sa%ings account accruing interest at a "=1*A rate= Houdeposit $our surplus o, C17*88 eac! mont! ,or a $ear e,ore purc!asing a !ouse= Hou appl$ ,or a "8$ear mortgage and get appro%ed ,or C**87888 at a 4="A interest rate= Hou are unsure i, $ou can a,,ord a!ouse t!at costs C**87888= 9se t!e Y> +ol%er to determine !o. muc! o, a mortgage $ou can a,,ordto tae out=

+a%ings 6ccount Y>

>ortgage Y>

N=

I%=

PV=

PMT=FV=

P/Y=

C/Y=

PMT: END BEGIN

N=

I%=

PV=

PMT=

FV=

P/Y=

C/Y=

PMT: END BEGIN

N=

I%=

PV=

PMT=

FV=

P/Y=

C/Y=

PMT: END BEGIN

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Unit- Exponential and Log Functions

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