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The Use and Abuse of Logarithmic Axes 

HarveyJ.Motulskyhmotulsky@graphpad.com

©2009,GraphPadSoftware,Inc.June2009

Mostgraphingprogramscanplotlogarithmicaxes,whicharecommonlyusedandabused.Thisarticleexplainstheprinciplesbehindlogarithmicaxes,soyoucanmakewisechoicesaboutwhentousethemandwhentoavoidthem.

What is a logarithmic axis? 

A logarithmic axis changes the scale of an axis.  

Thetwographsbelowshowthesametwodatasets,plottedondifferentaxes.

Thegraphonthelefthasalinear(ordinary)axis.Thedifferencebetweeneverypairofticksisconsistent(2000inthisexample).

Thegraphontherighthasalogarithmicaxis.Thedifferencebetweeneverypairofticksisnotconsistent.Fromthebottomtick(0.1)tothenexttickisadifferenceof0.9.Fromthetoptick(100,000)downtothenexthighesttick(10,000)isa

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differenceof90,000).Whatisconsistentistheratio.Eachaxistickrepresentsavaluetenfoldhigherthantheprevioustick.

Thereddotsplotadatasetwithequallyspacedvalues.EachdotrepresentsavaluewithaYvalue500higherthanthedotbelow.Thedotsareequallyspacedonthegraphontheleft,butfarfromequallyspacedonthegraphontheright.Topreventoverlap,thepointsarejitteredtotherightandleftsotheydon'toverlap.Thehorizontalpositionofthereddotshasnoothermeaning.

ThebluedotsrepresentadatasetwhereeachvaluerepresentsaYvalue1.5timeshigherthantheonebelow.Onthegraphontheleft,thelowervaluesarealmostsuperimposed,makingitveryhardtoseethedistributionofvalues(evenwithhorizontaljittering).Onthegraphontherightwithalogarithmicaxis,thepointsappearequallyspaced.

Interpolating between log ticks 

Whatvalueishalfwaybetweenthetickfor10andtheonefor100?Yourfirstguessmightbetheaverageofthosetwovalues,55.Butthatiswrong.Valuesarenotequallyspacedonalogarithmicaxis.Thelogarithmof10is1.0,andthelogarithmof100is2.0,sothelogarithmofthemidpointis1.5.Whatvaluehasalogarithmof1.5?Theansweris101.5,whichis31.62.Sothevaluehalfwaybetween10and100onalogarithmicaxisis31.62.Similarly,thevaluehalfwaybetween100and1000onalogarithmicaxisis316.2.

Why “logarithmic”?  

Intheexampleabove,theticksat1,10,100,1000areequallyspacedonthegraph.Thelogarithmsof1,10,100and1000are0,1,2,3,whichareequallyspacedvalues.Sincevaluesthatareequallyspacedonthegraphhavelogarithmsthatareequallyspacednumerically,thiskindofaxisiscalleda“logarithmicaxis”.

Lingo 

Thetermsemilogisusedtorefertoagraphwhereoneaxisislogarithmicandtheotherisn’t.Whenbothaxesarelogarithmic,thegraphiscalledalog­log plot.

Other Bases 

Allthelogarithmsshownabovearecalledbase10logarithms,becausethecomputationstake10tosomepower.Thesearealsocalledcommonlogarithms.

Mathematiciansprefernaturallogarithms,usingbasee(2.7183...).Buttheydon’tseemverynaturaltoscientists,andarerarelyusedasanaxisscale.

Abase2logarithmisthenumberofdoublingsittakestoreachavalue,andthisiswidelyusedbycellbiologistsandimmunologists.Ifyoustartwith1anddoubleitfourtimes(2,4,8,and16),theresultis16,sothelogbase2of16is4.

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GraphPadPrismcanplotlog2axes.ChoosefromtheupperleftcorneroftheFormatAxisdialog.

Logarithmic axes cannot contain zero or negative numbers 

The logarithms of negative numbers and zero are simply not defined 

Let’sstartwiththefundamentaldefinitionofalogarithm.If10L=Z,thenListhelogarithm(base10)ofZ.IfLisanegativevalue,thenZisapositivefractionlessthan1.0.IfLiszero,thenZequals1.0.IfLisgreaterthan0,thenZisgreaterthan1.0.NotethattherenovalueofLwillresultinavalueofZthatiszeroornegative.Logarithmsaresimplynotdefinedforzeroornegativenumbers.

Thereforealogarithmicaxiscanonlyplotpositivevalues.Theresimplyisnowaytoputnegativevaluesorzeroonalogarithmicaxis.

A trick to plot zero on a logarithmic axis in Prism 

Ifyoureallywanttoincludezeroonalogarithmicaxis,you’llneedtobeclever.Don’tenter0,insteadenterasasmallnumber.Forexample,ifthesmallestvalueinyourdatais0.01,enterthezerovalueas0.001.ThenusetheFormatAxisdialogtocreateadiscontinuousaxis,andusetheAdditionalticksfeatureofPrismtolabelthatspotontheaxisas0.0.

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Logarithmic axes on bar graphs are misleading 

Thewholepointofabargraphisthattherelativeheightofthebarstellsyouabouttherelativevaluesplotted.Inthegraphbelow,onebarisfourtimesastallastheotherandonevalue(800)isfourtimestheother(200).

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Sincezerocan’tbeshownonalogaxis,thechoiceofastartingplaceisarbitrary.Accordingly,therelativeheightoftwobarsisnotdirectlyrelatedtotheirrelativevalues.Thegraphsbelowshowthesamedataasthegraphabove,butwiththreedifferentchoicesforwheretheaxisbegins.Thisarbitrarychoiceinfluencestherelativeheightofthetwobars,amplifiedinthegraphontheleftandminimizedinthegraphontheright.

Ifthegoalistocreatepropaganda,abargraphusingalogarithmicaxisisagreattool,asitletsyoueitherexaggeratedifferencesbetweengroupsorminimizethem.Allyouhavetodoiscarefullychoosetherangeofyouraxis.Don’tcreatebargraphsusingalogarithmicaxisifyourgoalistohonestlyshowthedata.

When to use a logarithmic axis 

A logarithmic X axis is useful when the X values are logarithmically spaced 

TheX‐axisusuallyplotstheindependentvariable–thevariableyoucontrol.IfyouchoseXvaluesthatareconstantratios,ratherthanconstantdifferences,thegraphwillbeeasiertoviewonalogarithmicaxis.

Thetwographsbelowshowthesamedata.Xisdose,andYisresponse.Thedoseswerechosensoeachdoseistwicethepreviousdose.Whenplottedwithalinearaxis(left)manyofthevaluesaresuperimposedanditishardtoseewhat’sgoingon.

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Withalogarithmicaxis(right),thevaluesareequallyspacedhorizontally,makingthegrapheasiertounderstand.

A logarithmic axis is useful for plotting ratios 

Ratiosareintrinsicallyasymmetrical.Aratioof1.0meansnochange.Alldecreasesareexpressedasratiosbetween0.0and1.0,whileallincreasesareexpressedasratiosgreaterthan1.0(withnoupperlimit).

Onalogscale,incontrast,ratiosaresymmetrical.Aratioof1.0(nochange)ishalfwaybetweenaratioof0.5(halftherisk)andaratioof2.0(twicetherisk).Thus,plottingratiosonalogscale(asshownbelow)makesthemeasiertointerpret.Thegraphbelowplotsoddsratiosfromthreecase‐controlretrospectivestudies,butanyratiocanbenefitfrombeingplottedonalogaxis.

Thegraphabovewascreatedwiththeoutcome(theoddsratio)plottedhorizontally.Sincethesevaluesaresomethingthatwasdetermined(notsomethingsetbytheexperimenter),thehorizontalaxisis,essentially,theY‐axiseventhoughitishorizontal.

A logarithmic axis linearizes compound interest and exponential growth 

Thegraphsbelowplotexponentialgrowth,whichisequivalenttocompoundinterest.Attime=0.0,theYvalueequals100.Foreachincrementof1.0ontheX

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axis,thevalueplottedontheYaxisequals1.1timesthepriorvalue.Thisisthepatternofcellgrowth(withplentyofspaceandnutrients),andisalsothepatternbywhichaninvestment(ordebt)growsovertimewithaconstantinterestrate.

Thegraphontherightisidenticaltotheoneontheleft,exceptthattheYaxishasalogarithmicscale.Onthisscale,exponentialgrowthappearsasastraightline.

Exponentialgrowthhasaconstantdoublingtime.Forthisexample,theYvalue(cellcount,valueofinvestment…)doublesforeverytimeincrementof7.2657.

An exponential decay curve is linear on a logarithmic axis, but only when it decays to zero 

Thegraphsbelowshowexponentialdecaydowntoabaselineofzero.Thiscouldrepresentradioactivedecay,drugmetabolism,ordissociationofadrugfromareceptor.

Thehalf‐lifeisconsistent.Forthisexample,thehalf‐lifeis3.5minutes.Thatmeansthatattime=3.5minutes,thesignalishalfwhatitwasattimezero.Attime=7.0minutes,thesignalhascutinhalfagain,to25%oftheoriginal.Bytime=10.5minutes,ithasdecayedagaintohalfwhatitwasattime=7.5minutes,whichisdownto12.5%oftheoriginalvalue.

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Notethatwhenanexponentialdissociationcurveplateausatavalueotherthanzero,itwillnotbelinearonalogarithmicaxis.ThegraphontherightbelowhasalogarithmicYaxis,buttheexponentialdecay(greencurve)isnotastraightline.

Lognormal distributions  

Plotting lognormal distributions on a logarithmic axis 

Thetwographsbelowplotthesame50values.Thegraphonthelefthasalinear(ordinary)Yaxis,whilethegraphontherighthasalogarithmicscale.

Thedataaresampledfromalognormaldistribution,whichisveryasymmetrical.

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Whenshownonalinearscale(graphontheleft),itisimpossibletoreallygetasenseofthedistribution,sinceabouthalfofthevaluesareplottedinpileatthebottomofthegraph.Anotherproblemisthatifyousawonlythegraphontheleft,youmightthinkthehighestfourvaluesareoutliers,sincetheyseemtobesofarfromtheothers.

Whenplottedwithalogarithmicaxis,thedistributionappearssymmetrical,thehighestpointsdon’tseemoutofplace,andyoucanseeallthepoints.

Thesedatacomefromalognormaldistribution.ThatmeansthatthelogarithmsofvaluesfollowaGaussiandistribution.Plottingsuchvaluesonalogarithmicplotmakesthedistributionmoresymmetricalandeasiertounderstand.

The mean and geometric mean  

Thegraphsbelowshowthesamedata,andalsoshowthemeanandgeometricmean.Thegeometricmeaniscomputedbyfirsttransformingallthevaluestologarithms,findingthemeanofthoselogarithms,andthenreversetransformingthatmeanbacktotheoriginalunits.

Withanasymmetricaldistribution,themeanandgeometricmeancanbequitedifferentasshownhere.Whenplottedonalogscale(right),themeanisnotnearthemiddleofthedata.Thatisbecausethepointsabovethemeanhavemuchlargervaluessobringupthemeanvalue.Onalogarithmicaxis,thegeometricmeanisnearthemiddleofthedistribution.

Thisgraphshowsthattheconceptofanaverageormeanissomewhatambiguouswhendataareplottedonalogarithmicaxis.Doyoumeanthemeanoftheactualvalues,orthemeanofthelogarithms(thegeometricmean)?Becauseofthis

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ambiguity,considershowingthemedianinstead.Forthesedata,themeanis1541.3,thegeometricmeanis141.2,andthemedianis116.3

Displaying variability on a lognormal distributions 

Intheexampleabove,thestandarddeviation(SD)is4930.TherangethatcoversameanplusorminusoneSD,therefore,extendsfrom1541–4930to1541+4930,whichisfrom‐3389to6741.Thisrangecannotbeplottedonalogarithmicaxis,becausesuchanaxiscannotincludenegativevalues

Onealternativeistocomputeageometric standard deviation,butthesearenotoftenused(muchlessthangeometricmeans)andaretrickytounderstand.

Fordisplaypurposes,ifyouarenotgoingtographeveryindividualvalueasonthegraphsabove,abox‐and‐whiskersplotasshownbelowdoesagreatjobofshowingvariation.Theboxextendsfromthe25thto75thpercentiles,withalineatthemedian(50thpercentile).Thewhiskers,onthisgraph,godowntothesmallestvalueanduptothelargestvalue(alternativedefinitionsareoftenused).Again,thegraphiseasytounderstandwhenalogarithmicaxisisused(right)butisnotveryhelpfulwithalinearaxis.

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Distinguish using a logarithmic axis from plotting logarithms 

Regression fits the data, not the graph 

Becarefulwhenyoufitacurvetodatawithalogarithmicaxis.Whenyoufitacurvewithnonlinearregression(oralinewithlinearregression),youfitamodel(equation)thatdefinesYasafunctionofX.Choosingtostretcheitheraxistoalogarithmicscaledoesnotchangeanyvalues.

Thetwographsbelowplotthesamedata.Thecurveintheleftgraphwascreatedbyusingnonlinearregressiontofitanexponentialdecaycurve.Thatcurveisalsoshownonthegraphontheright.Withalogarithmicaxis,anexponentialcurvedecayingtozerolooksstraight.

 

Sincethegraphontherightlookslikeastraightline,youmightbetemptedtofitthedatawithlinearregression,ratherthanfittinganexponentialdecaymodelusingnonlinearregression.Thegraphsbelowshowswhathappenswhenyoufitlinearregressiontothosedata.Onalinearaxis(left),thelinearregressionlineindeedlookslikealine.Onalogarithmicaxisincontrast(right),thelinearregressionlineappearscurved.Linearregression(andnonlinearregression)fitthedata,notthegraph.NotethatfourpointshavenegativeYvalues,andthesearesimplyomittedfromthegraphontheright.

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Use antilog or powers‐of‐ten numbering when plotting values that are logs 

Prism’sbuilt‐indose‐responseequationsallassumethattheXvaluesarelogarithmsofdosesorconcentrations.EnteringtheXvaluesasconcentrationsordoses,andthenstretchingtheaxistoalogarithmicscaleisnotthesameatall.ThatapproachdoesnotchangetheXvalues.

ImaginethattheXvaluesrangefrom1to10,000,andyouwanttofitadose‐responsecurveusingPrism’sbuilt‐inequations.Don’tusealogarithmicaxis.Instead,transformthevaluestologarithms(eitherbeforeenteringthedata,orusingPrism’sTransformanalysis),andplotthese(whicharelogarithms)onalinearaxis.AftertransformingtheXvalueswillrangefrom0to5,andtheXaxiswilllooklikethetopexamplebelow.

Prismletsyouformattheaxistoshowtheoriginalvalues(doses)ratherthanthetransformedvalues(logarithmsofdoses).Thethreeaxesaboveallhavethesame

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range(0to5).Thefirstaxishasdecimalnumbering,thesecondaxishasantilognumbering,andthethirdaxisusespowers‐of‐tennumbering.ThescreenshotshowshowtoselecttheseontheFormatAxisdialog.

Notethatthebottomtwoaxeshavenotbeenstretchedtoalogarithmscale.Onlythenumbershavebeenwritteninalternativeformats.Eventhoughthetopoftheaxisislabeled10,000or105,thecorrespondingvalueonthegraphactuallyisY=5.AvalueofY=10,000wouldbewayoffscale,unlessyoutransformeditfirst.