082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

92 Chapter 2 Engineering functions
x
1 1
2 64
3 729
4 4096
5 15625
6 46656
7 117649
8 262144
9 531441
10 1000000
y
log y
6
5
4
3
2
1
0
1
2 3 4 5 6 7 8 9 10 x
Figure2.33
Thefunctiony =x 6 plotted on alog--linear graph.
Useoflog–logandlog–linearscales
Suppose wewish toplot
y(x)=x 6
1x10
This may appear a straightforward exercise but consider the variation in the x and y
values. Asxvaries from 1 to 10, thenyvaries from 1 to 1000000, as tabulated above.
Severalofthesepointswouldnotbediscernibleonagraphandsoinformationwouldbe
lost.Thiscanbeovercomebyusingalogscalewhichaccommodatesthelargevariation
iny. Thus log yisplotted againstx, rather thanyagainstx. Note thatinthisexample
logy=logx 6 =6logx
so asxvaries from 1 to 10, logy varies from 0 to 6. A plot in which one scale is logarithmic
and the other is linear is known as a log--linear graph. Figure 2.33 shows logy
plottedagainstx.Ineffect,useofthelogscalehascompressedalargevariationintoone
which ismuch smaller and easier toobserve.
Example2.16 Considery = 7 x for −3 x 3.Plot a log--linear graph of this function.
Solution We have
and so
y=7 x
logy = log(7 x ) =xlog7 = 0.8451x
PuttingY = logy we haveY = 0.8451x which is the equation of a straight line passing
through the origin with gradient log7. Hence when logy is plotted againstxastraight
linegraphisproduced.ThisisshowninFigure2.34.Notethatbytakinglogs,therange
on the verticalaxis has been greatly reduced.
Aplotinwhichbothscalesarelogarithmicisknownasalog--logplot.Herelogyis
plotted against logx.