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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