USINGLOGRARITHMICGRAPHPAPER

Logarithmic graph paper is seldom seen by students in elementary courses. Thereseemstobeageneralimpressionthat(1)itistoodifficulttodealwith,or(2)students"canpickitup"withoutinstructions.Nietheristrue,inmyopinion.

Inthedaysofsliderules,studentshad(oroughttohavehad)intimatefamiliaritywithlogrithms and logarithmic scales, for every slide rule had at least two such scales.Nowadays many of these gory details are hidden in the innards of an electroniccalculator or computer, a "black box" that grinds out numbers,whether or not thosenumbershaveanysignificance.

Graphs with logarithmic scales are found in research papers and textbooks. Oneexampleistheusualgraphoftheelectromagneticspectrum.Ifstudentsaretointerpretsuchgraphsintelligently,theyneedtodirectlyexperiencetheprocessofmakingone.

All sorts of computer graphing software is available. The most used software isdesignedfortheneedsofbusiness,notscience.Manysuchsoftwarepackagessimplycannotdothethingsnecessaryfordealingwiththeneedsofphysics.

7.8LOGARITHMICGRAPHPAPER

Logarithmic graph papers are available inmany types. They simplify the process oflinearizing exponential and power relations and determining the constants in theirequation.

Fig.7.8.Relabelingalogarithmicscale.

Examine the logarithmic scalesof suchpaper.Typically the logarithmic scales comewithpartiallylabeleddivisions.AsampleisshowninFig.7.8.

Startingfromtheleft,weseelabelsfrom1to10,repeatedthreetimes.Eachsegmentlabeledfrom1to10iscalledacycle.Thissamplehas3cycles.Thelabelingalreadyprintedonthepaperisforconvenienceonlytheusermustrelabeltheaxestosuitthedata.

Forexample,supposeyouwanttoplotdatavaluesrangingfrom2unitsto800unitson

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thisthreecyclelogscale.Thefirstcycleislabeled"asis"from1to9.Thesecondcycleisrelabeled10,20,30,40,50,60,70,80,and90.Thethirdcycleisrelabeled100,200,etc.to1000.

Eachcyclecoversarangeofvaluesspanningonefactorof10.Thenextcyclecoversarange10timeslarger,etc.Thevaluezerodoesnotappearona logarithmicscale,forlog(0)=minusinfinity.

The"generic"labelingisdonedifferentlyondifferentbrandsofpaper.Somelabelthestartofacyclewith"1"someuse"10".Somemark1.5and2.5somedon't.

The user has less freedom in labeling a log scale than a linear one. The scales aredirected,i.e.,oneway,andcannotbelabeledintheoppositedirection.

Theunderlyingprincipleofthelogscaleisthatlengthsalongthescaleareproportionaltothelogarithmsoftheplottedvalues.TheCandDscalesofsliderulesareconstructedthesameway.

The user must carefully examine themarkings of the scale to correctly interpret itssubdivisions.Someintervalshave10subdivisions,someonlyfive(everysecondoneofthetenbeingshown).Errorsintheuseoflogpaperresultfromfailuretonoticethesedifferencesinthewaysubdivisionsarelabeled.

Graph paperswith one linear and one logarithmic scale are called semilogarithmic,loglinear,orsimplylogarithmic.Graphpaperswithbothscaleslogarithmicarecalledloglog,fulllog,orduallogarithmic.

As is evident from the 10 divisions of each cycle, these logarithmic scales representbase10logarithms.

EXPONENTIALRELATIONSoftheform

y=Aex

whereAandareconstant,canberenderedasstraightlinesonsemilogpaper.Toseethis,takethelogarithmofbothsidesof

y/A=ex

whichgives

logylogA=xloge=x(0.43420...)

Plotxonthelinearaxisandyonthelogarithmicaxis.Theresultingstraightlinewillhaveslope

logy2logy1=loge

Fig.7.9.Semilogarithmicgraphpaper.

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x2x1

fromwhichtheconstantmaybeeasilydetermined.

TheyinterceptofthegraphwillbethevalueoftheconstantA.

The slope of a line on logarithmic paper must be interpreted with care. It may beevaluatedineitheroftwoways.

(1)Choose twowellseparated points on the line (x1,y1) and (x2,y2). Use these toevaluatetheleftsideofequation(715).

(2) The value of the numerator of equation (715) may be found by lengthmeasurementsonthegraph,sincethelogarithmsareproportionaltolengths.Measurethelengthofonecycleonthepaper.Measurethelengthbetweeny2andy1.Then

L/C=log2ylog1y

whereListhelengthalongthelogaxisbetweenthetwopoints,andCisthelengthofonecyclemeasuredalongthataxis.

Thisistruebecauseforonecycle,

log(10y)log(y)=log(10)=1.

Thismethodmakes it unnecessary to look up or evaluate the logarithms of the datapoints.

Example: This graph happens to have cycles 4.15 cmlong. [Because of unavoidable size changes in thereproduction process, themeasurementswehavemadeon the original will be different from those on yourcopy. Measurements made from your copy will,howevergivethesamefinalvaluesofAand.]

Thevariablexisplottedonthehorizontalaxisandyisplottedontheverticalaxis.

Whenx=0,theny=Ae0=A.SotheyinterceptisthevalueofA.Butx=0liesofftheedgeofthegraph,sothiswouldnotbeagoodwaytofindthevalueofAinthiscase.Letuspostponethisproblemandfirstfindtheslope.

Choose two points on the line. I chose the points with values x=2 and x=7 on thehorizontalaxis.MarkthesepointsonyourcopyofFig.7.8.Theyhavevaluesofy=20andy=500ontheverticalaxis.Arulershowsthat20and500areseparatedby5.85cm

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vertically.

Theslopeofthelineis

5.85cm4.15cm1.41==0.2819=loge725

andtherefore,sincelog(e)=0.4342945

0.2819==0.64910.4343

Thenumeratoron the leftofEq.718 isdimensionless, so theunitsofwillbe thereciprocaloftheunitsofx.

NowwecanevaluatetheconstantA.Takeanyvalueofxthatisonthegraph,sayx=7.Fromthegraph,thisgivesy=500.Then

y500500A====5.317x794.04ee

Sofinally,theequationthatrepresentsthiscurveis

y=5.317e(0.6491)x

Noticethatinthiswholeprocessweneverhadtotakealogarithmofanydatavalues.

EXERCISE(7.8)CheckEq.721byevaluatingyforx=3.4andcomparingtheresultwiththegraph.{Answer48.32theagreementisgood.}

EXERCISE(7.9)UseEq.721tocalculateyforx=6.2.Comparewiththegraph.

POWERRELATIONSoftheform

y=Kxp

mayberenderedstraightbyplottingonloglogpaper.Takethelogarithmofbothsides

logy=logK+plogx

Nowifyisplottedagainstxonloglogpaperwegetalineofslope

logy2logy1=plogx2logx1

where, as before, the points 1 and 2 are wellseparated points on the straight line.Neitherintercepthasanyspecial,useful,meaningon thisgraph! (On logpaper logxneverhasazerovalue.)

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Fig.7.10.Logloggraphpaper.

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We have several ways to calculate the slope, once we have specified two wellseparatedpointsontheline,andobtainedequation(724).

(1)Evaluateequation(724)byexplicitcalculationofthelogarithms.

(2)Setuptherighttriangleandmeasurethelengthsofthelegs.Then,asbefore:

lengthofthe"y"leglengthofone"y"cycle =plengthofthe"x"leg lengthofone"x"cycle

Onmostlogloggraphpaper,thelengthsofthecyclesonthexandyaxesaresimplyrelated.Iftheyareequal,thentheslopeissimply

lengthofone"y"leg=tan=plengthofthe"x"leg

Sointhisspecialcaseonemaysimplymeasuretheangle,,todeterminetheslope.

Example:Datathatplotsasastraightline on loglog paper can always beexpressedbytherelation

y=Kxp

The constants K and p are to bedetermined from the graph. Twopoints have been chosen in Fig. 7.10andflaggedwithcircles.Thesehappentohaveconvenientvaluesandarewellseparatedontheline.Theyare(2,300)and(8,40).

The procedure is simplified herebecausethecyclesarethesamelengthon both axes. Therefore the slope issimply the length ratio of the legs oftherighttriangleThatis:

y4.42cm==0.5423=px8.15cm

Asa check, the slope angle,measuredwith aprotractor, is 28, approximately. Thetangentofthisis0.5317.Consideringmeasurementerrorsonthissmallgraph,thisisgoodagreement.Inyourexperimentalwork,alwaysusethelargestpossibleareaofthegraphpaper.

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Therefore:

y=Kx0.5423

Nowwe can determineK be taking any point on the line and solving this equation.Let'stakethepointx=2.Atthispointy=300.

x0.5423=0.6867

yK==300/0.6867=436.9x0.5423

Sotheequationofthislineis:

y=436.90.5423x

Thisexpressionshowsmoredigits than is justifiedby theprecisionofmeasurementsfrom this size of graph. An examination of the actual data uncertainties would helpdeterminetheappropriateamountofroundingforKandp.

EXERCISE 7.10: Check Eq. 731 by calculating y at x=80 and compare with thegraph.{Answer:40.58}

EXERCISE 7.11: Check Eq. 731 by calculating y at x=29 and compare with thegraph.

EXERCISE7.12:UseEq.731tocalculatexfory=70.Comparewiththegraph.

OTHER RELATIONS. The above discussion covers only the most commonlyencounteredcases.Otherspecialgraphpapersareavailable,whichcanstraightenoutgraphsofmanykindsofrelations.Tolistafewpolarcoordinate,bipolarcoordinate,elliptical,hyperbolic,Smithcharts(agridoforthogonalcircles).Ithasbeensuggestedas a joke that someone ought to print graph paper on rubber sheets, so you couldstraightenoutanycurvebywarpingthegridlines!Butthatiswhatthesevariouspapersdo for you. They are only useful if the warping of the grid lines corresponds to anaccuratelyknownmathematical relation thatcanbemathematically transformedbacktoalineargrid.

RESCALINGLOGARITHMICSCALES.Wehavesofaronlyconsideredrescalinglogscalesbymultiplyingeachscalevaluebysome factorof10.Thispreserves the cycle length.Otherlabelingsarepossible,butonemustexercisegreatcaretoavoidblundersinplotting.

(1) The scale values may be multiplied by any commonfactor.Afactorof2or4iseasiesttodealwith.Thisoperationpreserves cycle length, but shifts the cycles along the axis.Thiscanbeusefulwhenthedataspansonlyonefactorof10,

Fig.7.13.Slopeonduallogpaper.

Fig.7.11. Fig.7.12.

butwouldfallintwoadjacentdecades.SeeFig.7.11.

(2) The scale values may be raised to any common power.Thisexpands or contracts the cycle size.This can be usefulwhen you need two cycle paper but have only one cyclepaper. Raise all of the values printed along the scale to thesecond power. See Fig. 7.12. This should be consideredstrictlyanemergencyexpedient,neveracceptable inagraphintendedforpublication.

Anotheremergency trick is tosplice thepaper.Supposeyouhaddatathatspannedonlyonefactorof10butspannedtwocyclesofthepaper.Forexample,thevaluesmightrangefrom45 to 312. If you used two cycle paper, this graph wouldoccupylessthanhalfofthepaper.Ifyoucutonecyclepaperat the "4"mark, it couldbe spliced together so that it readsfrom40to400.

SLOPES ON LOGLOG PAPER. Itsometimeshappensthat thepowervalueinanequation isknownorassumed thepurposeofthe graphical analysis is to find some otherparameteroftheequation.Itisthenjustifiableto impose a straight line of the known slopeonto the data points. It is easiest to constructlinesofslopecorrespondingto integerpowersorreciprocalsofintegers.Supposeyouwantedaslopeforpower2.Markthepoint(n2,n)andconnectittothepoint(1,1)withastraightline.The number n can be any convenient value.Supposewe taken=2.Thenmark thepoints(4,2)and(1,1)andconnectthemwithastraightline,whichwillhaveslope2.

Ifyouwantedthelinecorrespondingtoapower3,youmighttaken=2,thenn3=8.Soconnectthepoints(8,2)and(1,1)withastraightline,whichwillhaveslope3.

Fig.7.13showsthis,forseveraldifferentvaluesofslope.

1999,2004,byDonaldE.Simanek.