Graphs on Logarithmic and Semilogarithmic Paper


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1 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl dt nd determine n pproximte eqution describing the dt. In this short chpter we supplement our other grphicl methods b introducing other kinds of grph pper, the logrithmic nd semilogrithmic. We will see tht grphs mde on these kinds of pper enble us to deduce things bout dt tht were not evident when plotted on ordinr rectngulr coordinte pper. In prticulr, it will often enble us to find n eqution to describe tht dt, process known s curve fitting.
2 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper Logrithmic nd Semilogrithmic Pper FIGURE Logrithmic grph pper. Our grphing so fr hs ll been done on ordinr grph pper, on which the lines re equll spced. For some purposes, though, it is better to use logrithmic pper (Fig. ), lso clled loglog pper, or semilogrithmic pper (Fig. ), lso clled semilog pper. Looking t the logrithmic scles of these grphs, we note the following:. The lines re not equll spced. The distnce in inches from, s, to, is equl to the distnce from to, which, in turn, is equl to the distnce from to.. Ech tenfold increse in the scle, s, from to 0 or from 0 to 00, is clled ccle. Ech ccle requires the sme distnce in inches long the scle. 3. The log scles do not include zero. Looking t Fig., notice tht lthough the numbers on the verticl scle re in equl increments (,, 3,..., 0), the spcing on tht scle is proportionl to the logrithms of those numbers. So the numerl is plced t position corresponding to log (which is 0.0, or bout onethird of the distnce long the verticl); is plced t log (bout 0. of the w); nd 0 is t log 0 (which equls, t the top of the scle). When to Use Logrithmic or Semilog Pper We use these specil ppers when:. The rnge of the vribles is too lrge for ordinr pper.. We wnt to grph power function or n exponentil function. Ech of these will plot s stright line on the pproprite pper, s shown in Fig We wnt to find n eqution tht will pproximtel represent set of empiricl dt. Grphing the Power Function A power function is one whose eqution is of the following form: FIGURE pper. Semilogrithmic grph Power Function x n where nd n re nonzero constnts. This eqution is nonliner (except when n ), nd the shpe of its grph depends upon whether n is positive or negtive nd whether n is greter thn or less thn. Figure 3 shows the shpes tht this curve cn hve for vrious rnges of n. If we tke the logrithm of both sides of Eq., we get log log(x n ) log n log x If we now mke the substitution X log x nd Y log, our eqution becomes Y nx log This eqution is liner nd, on rectngulr grph pper, grphs s stright line with slope of n nd intercept of log (Fig. ). However, we do not hve to mke the substitutions shown bove if we use logrithmic pper, where the scles re proportionl to the logrithms of the vribles x nd. We simpl hve to plot the originl eqution on loglog pper, nd it will be stright line which hs slope n nd which hs vlue of when x (see Fig. 3). This will be illustrted in the following exmple.
3 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge 3 Section Logrithmic nd Semilogrithmic Pper 3 On ordinr grph pper On logrithmic or semilog grph pper Eq. = x n where 0 < n < Slope = n 0 x x Power function Eq. = x n where n > 0 x Slope = n x On logrithmic grph pper Eq. = x n where n < 0 Slope = n 0 x x Exponentil function Eq. = (b) nx Exponentil growth Text Eq. = e nt where n > 0 Exponentil dec Text Eq. = e nt where n < 0 0 x 0 t Slope = n log b 0 x Slope = n log e 0 t Slope = n log e On semilog grph pper 0 t 0 t FIGURE 3 The power function nd the exponentil function, grphed on ordinr pper nd loglog or semilog pper. Exmple : Plot the eqution.x. for vlues of x from to 0. Choose grph pper so tht the eqution plots s stright line. Solution: We mke tble of point pirs. Since the grph will be stright line, we need onl two points, with third s check. Here, we will plot four points to show tht ll points do lie on stright line. We choose vlues of x nd for ech compute the vlue of. x Y Slope = n intercept = log X FIGURE Grph of Y nx log on rectngulr grph pper.
4 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper We choose loglog rther thn semilog pper becuse we re grphing power function, which plots s stright line on this pper (see Fig. 3). We choose the number of ccles for ech scle b looking t the rnge of vlues for x nd. Note tht the logrithmic scles do not contin zero, so we cnnot plot the point (0, 0). Thus on the x xis we need one ccle. On the xis we must go from. to.. With two ccles we cn spn rnge of to 00. Thus we need loglog pper, one ccle b two ccles. We mrk the scles, plot the points s shown in Fig., nd get stright line s expected. We note tht the vlue of t x is equl to. nd is the sme s the coefficient of x. in the given eqution (0,.) (,.). Slope = x FIGURE Grph of.x.. We cn get the slope of the stright line b mesuring the rise nd run with scle nd dividing rise b run. Or we cn use the vlues from the grph. But since the spcing on the grph rell tells the logrithm vlue of the position of the pictured points, we must remember to tke the logrithm of those vlues. (Either common or nturl will give the sme result.) Thus slope r is ru e n ln. ln. ln ln. The slope of the line is thus equl to the power of x, s expected from Fig.. We will use these ides lter when we tr to write n eqution to fit set of dt. Common Error Be sure to tke the logs of the vlues on the x nd xes when computing the slope.
5 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Section Logrithmic nd Semilogrithmic Pper Grphing the Exponentil Function Consider the exponentil function given b the following eqution: Exponentil Function (b) nx If we tke the logrithm of both sides, we get log log b nx log (n log b)x log If we replce log with Y, we get the liner eqution Y (n log b) x log If we grph the given eqution on semilog pper with the logrithmic scle long the xis, we get stright line with slope of n log b which cuts the xis t (Fig. 3). Also shown re the specil cses where the bse b is equl to e (....). Here, the independent vrible is shown s t, becuse exponentil growth nd dec re usull functions of time. Exmple : Plot the exponentil function 00e 0.x for vlues of x from 0 to 0. Solution: We mke tble of point pirs. x We choose semilog pper for grphing the exponentil function nd use the liner scle for x. The rnge of is from 3. to 00; thus we need one ccle of the logrithmic scle. The grph is shown in Fig.. Note tht the line obtined hs (0, 00) 0 Slope = 0. 0 (0, 3.) x FIGURE Grph of 00e 0.x.
6 0CH_PHClter_TMSETE_ //00 : PM Pge Grphs on Logrithmic nd Semilogrithmic Pper When computing the slope on semilog pper, we tke the logrithms of the vlues on the log scle, but not on the liner scle. Computing the slope using common logs, we get slope n log e log 3. log n log e s we got using nturl logs. The process of fitting n pproximte eqution to fit set of dt points is clled curve fitting. In sttistics it is referred to s regression. Here we will do onl some ver simple cses. intercept of 00. Also, the slope is equl to n log e or, if we use nturl logs, is equl to n. slope n ln 3. ln This is the coefficient of x in our given eqution. Empiricl Functions We choose logrithmic or semilog pper to plot set of empiricl dt when:. The rnge of vlues is too lrge for ordinr pper.. We suspect tht the reltion between our vribles m be power function or n exponentil function, nd we wnt to find tht function. We show the second cse b mens of n exmple. Exmple 3: A test of certin electronic device shows it to hve n output current i versus input voltge s shown in the following tble: (V) 3 i (A)..... Plot the given empiricl dt, nd tr to find n pproximte formul for in terms of x. Solution: We first mke grph on liner grph pper (Fig. ) nd get curve tht is concve upwrd. Compring its shpe with the curves in Fig. 3, we suspect tht the eqution of the curve (if we cn find one t ll) m be either power function or n exponentil function i (A) v (V) FIGURE Plot of tble of points on liner grph pper. Next, we mke plot on semilog pper (Fig. ) nd do not get stright line. However, plot on loglog pper (Fig. ) is liner. We thus ssume tht our eqution hs the form i n or ln i n ln ln In Exmple we showed how to compute the slope of the line to get the exponent n, nd we lso sw tht the coefficient ws the vlue of the function t x. Now
7 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Section Logrithmic nd Semilogrithmic Pper In Fig. below our dt plotted s nice stright line on logrithmic pper. But with rel dt we re often unble to drw stright line tht psses through ever point. The method of lest squres, not shown here, is often used to drw line tht is considered the best fit for scttering of dt points. i (A) FIGURE 3 v (V) Plot of tble of points on semilog pper. i (A) we show different method for finding nd n which cn be used even if we do not hve the vlue t x. We choose two points on the curve, s, (,.) nd (,.), nd substitute ech into getting nd ln i n ln ln ln. n ln ln 0 3 v (V) FIGURE Plot of tble of points on loglog pper. ln. n ln ln A simultneous solution, not shown, for n nd ields Our eqution is then n. nd. Here, gin, we could hve used common logrithms nd gotten the sme result. i.. Finll, we test this formul b computing vlues of i nd compring them with the originl dt, s shown in the following tble: 3 Originl i..... Clculted i..... We get vlues ver close to the originl. The fitting of power function to set of dt is clled power regression in sttistics. You m be ble to do this on our grphics clcultor. See problem of Exercise.
8 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper Exercise Grphs on Logrithmic nd Semilogrithmic Pper Grphing the Power Function Grph ech power function on loglog pper for x to 0.. x 3. 3x 3. x. x 3/. x. 3 x. /x. 3/x. x /3 Grph ech set of dt on loglog pper, determine the coefficients grphicll, nd write n pproximte eqution to fit the given dt. 0. x x Grphing the Exponentil Function Grph ech exponentil function on semilog pper.. 3 x 3. x. e x. e x. x/. 3 x/. 3e x/3. e x 0. e x/ You m be ble to do this on our grphics clcultor. See Problem. Grphing Empiricl Functions Grph ech set of dt on loglog or semilog pper, determine the coefficients grphicll, nd write n pproximte eqution to fit the given dt.. x x Current in tungsten lmp, i, for vrious voltges, : (V) i (ma) Difference in temperture, T, between cooling bod nd its surroundings t vrious times, t: t (s) T ( F)
9 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Section Logrithmic nd Semilogrithmic Pper. Pressure, p, of lb of sturted stem t vrious volumes, : (ft 3 ) p (lb/in. ) Mximum height reched b long pendulum t seconds fter being set in motion: t (s) 0 3 (in.) Grphics Clcultor. Some grphics clcultors cn fit stright line, power function, n exponentil function, or logrithmic function to given set of dt points, nd cn give the two constnts in the function. Such fitting goes b the sttistics nme of regression. The TI, for exmple, cn do liner, logrithmic, exponentil, nd power regression. You must enter the dt points nd then choose the tpe of function tht ou think will fit. The clcultor will give the two constnts. It will lso give the correltion coefficient, which is mesure of goodness of fit. If this coefficient is close to or, the fit is good; if it is close to 0, the fit is bd. Stud our clcultor mnul to lern how to do regression. Then use it for n of the problems through in this exercise set.
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