AN ALTERNATIVE TO THE LOAN LENGTH FORMULA

equation.

is a lesson on how the natu-

y = ex

ral log can be used in this particular case. The result-

To find the value of x when given a particular y, use the following alge-

ing formula is complex and

braic transformation.

again requires the proper placement of parentheses.

x = ln y

AN ALThTiEs iRs NreAadTaIsV"Ex e TqOua TlsHthEe nLaOtuAraNl l LoEgaNriGthTmHo fFyO"RorM"wUhLeAn (epis.189)

raised to the exponent x, the resulting value is y."

Before the use of calculators, people used a logarithm table to deter-

EXmAiMnePtLhEe2exponent values. Now, graphing calculators have a natural

CHECK YOUR UNDERSTANDING

Answer The length of the El7Xo.3aA5nMywePaorLuslEdif2dthroepmtoonatbholyut

Clloaguadreitwhmanktseyto[LbNor]r.oFwor$e2x5a,m00p0let,ocpounrscidhearsethaecfaor.llAowfteinr gloeoqkuinatgioant.his monthly1b3u0d=geetx, he realizes that all he can afford to pay per month is

$lo3Tea0otno0s.nogTelevhetede1f3tboo0ar,nbxyke,otsiuhsoaonttfhefieeasd,rtithntooegfiucanas5den.9ttshh%taeeyl[eoLwxaNpint]o.hknWieneynhhtaainttsodwbwuothduheglidncehtte?hnyeoteulrenn1ge3te0hdinotoftohratisihsee

ScOaLlUcuTlIaOtNor. TThoesorelvsuelthisisapprporbolxeimm,aitteilsyn4e.9ce. ssary to perform some

Sipnacyemtheentviasrianbcrlearseepdreb-y se$n5t0in. g time in the monthly paSytumdeennttsfonrmeeudlatoisreacnalcuexlaptoentehnet,tismoelvfionrgmfourlathfraotm exEpxoanmepnlter2equusiirnegs $th3e50 as ustheeomf loongtahrliythpmasy.mSeonmt.eAsk stsutduednetnstsmwayhaatlrtehaedyyehxpaveect

alTgheberdaeicvemloapnmipeunlattoiof nths eonloathnelemnognththfloyrmlouanla pisaybmeyeonntdfothrme usclao.pe of Tothfiisncdotuhresel.eTnhgtaht foofrtmhuellaoraenqugivreesntthheeuasme ofutnhteonf athtueraml olongtahrliythpmayi-n moerndte,rytoousonleveedfotor tsholevexfpoornthenetetx.ponent t. To solve for an exponent,

beweinll ihnatpropdeuncteodthtoelloegnsgtinh a proefvtihoeuslomanatbhecfooruerstheewyhaicletuffiorarslltoyitnhdteorrotsdh,uethcctiaisolmcnuatloayttibhoeentsh. eir

yLoesusonLneoe3ad-n6t,oLyueonnudgleetrhasrtnaFneoddrmatbhuoelucatotnhceepcot nosftaanntaet.uErxaalmliongeatrhiethfomllo. wIning

( ) ( ( ( ) )) eTbqoraufiiacntttid=orayntnh_l=.n_sef_eovx__Mrp_a_m1_lu2_a-_eltn_io_olf_nn1_x._+w__Mp__h_1__r-2e__n___1_gr_2_i_v_enwahpearreticMuprtla====r

monthly payment

principal

interest rate yn,uumsebtehreofoyleloawrsing

alge-

topic. This is not meant to be a comprehensive lesson on the use of logs. Rather, it is a lesson on how the natural log can be used in this particular case. The resulting formula is complex and again requires the proper

( ) ( ( ( ) )) TrBahSMeifsuiose=brdisest3titrotx0heu=0atetehd,uleanpasnees=ydx"o2prxf5o=ce,naq0e0lu0cn.a0u0tl,l5sax9tt,oh.trhese,npraeettsuo=urplatl_llein_nlu_og_sg_2_ev_a53_da_r,_0i_l0tau__00h__el0m__o1i_gs2_-aoly_nrf._i"ty_lh"n1_m_o+__2r_t_5__"30a__,_wb0._01_0__0lh0_2_5e__0_e_9_t_no-__ed__0ie__s.t1_0_e_2_5r__-9___

placement of parentheses.

CHECK YOUR UNDERSTANDING Answer The length of the

loan would drop to about

mCianlecutlhaeteetxoptohneennetavreaslut ehsu.nNdroewd,thgroafpahyineagr.calculattors h8.a9v6e a natural logarithm key [LN]. For example, consider the following equation.

Claude would need to take out a loan for about 9 years.

7.35 years if the monthly payment is increased by $50.

130 = ex

Students need to recalcu-

ToCsHoElvCe KforYxO, tUhRat Uis,NtoDfiEnRdSthTeAeNxpDoInNenGt to which you need to raise

ecaIotnlofc$uEg5xlea0attmo1hr3ap.0vlTe,ehy2oeo,nuwrethhnsuaeeltetldieminstgpoataphucpstorewoftxtohhiumeeld[alLotaNeanln]y?ikn4ec.y9re.aansed itnhethneemntoenrt1h3ly0 pinatyomtehnet

The development of the loan length formula is beyond the scope of

this course. That formula requires the use of the natural logarithm in order to solve for the exponent t.

late the time formula from Example 2 using $350 as the monthly payment. Ask students what they expect will happen to the length

of the loan before they actu-

ally do the calculations.

GraphiLnoga nCaLlecnugltahtoFro r (moru lGaraphing Software)

( ) (( ( ) )) t

=

_ln____Mp____-___l_n____Mp___-____1_r_2___

12ln

1

+

__r _ 12

where

M = m4o-3nthly paLyomaennCt alculations and Regression p = principal r = interest rate t = number of years

189

!"#$%&'!&()'!&*+%,-,+$./01122234(+'5+6"

'"7'87

Substitute p = 25,000, M = 300, and r = 0.059.

( ) (( ( ) ))

t

=

_ln____2__53__,_0_0__00__0__1_2_-l_n__ln1__+__2__5__30__,_0._01_0__00_2_5__0__9__-____0__.1_0__2_5 __9___

Calculate to the nearest hundredth of a year.

t 8.96

Claude would need to take out a loan for about 9 years.

CHECK YOUR UNDERSTANDING

In Example 2, what impact would an increase in the monthly payment of $50 have on the length of the loan?

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