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

2.4 Review of some common engineering functions and techniques 95
or more cycles. To decide which sort of graph paper is appropriate it is necessary to
examine the variation in size of the variable to be plotted measured in powers of 10. If,
for example,yvaries from 1 to 10, then paper with one cycle is appropriate. Ifyvaries
from 1 to 10 2 , two-cycle paper is necessary. If y varies from 10 −1 to 10 4 , then paper
with4 − (−1) = 5cycleswouldbeappropriate.Toseehowlog--linearpaperisusedin
practice, consider the following example.
Example2.18 During anexperiment the following pairs of data values wererecorded:
A B C D
x 0 1 5 12
y 4.00 5.20 14.85 93.19
It is believed thatyandxare related by the equationy = ab x . By plotting a log--linear
graph verify the relationship isof thisform and determineaandb.
Solution If the relationship isgiven byy =ab x , then taking logarithms yields
logy=loga+xlogb
So, plotting logy againstxshould produce a straight line graph with gradient logband
vertical intercept loga. The need to find logy is eliminated by plotting theyvalues directly
on a logarithmic scale. Examining the table of data we see that y varies from
approximately 10 0 to 10 2 so that two-cycle paper is appropriate. Values ofybetween 1
and 10 are plotted on the first cycle, and those between 10 and 100 are plotted on the
second.ThepointsareplottedinFigure2.37.Noteinparticularthatinthisexamplethe
‘1’atthestartofthesecondcyclerepresentsthevalue10,the‘2’representsthevalue20
and so on. From the graph, the straight line relationship between logy andxis evident.
It is therefore reasonable to assume that the relationship betweenyandxis of the form
y =ab x .
Tofindthegradientofthegraphwecanchooseanytwopointsontheline,forexample
Cand B. The gradient isthen
log14.85 −log5.20
5 −1
= log(14.85/5.20)
4
Recall thatlogbisthe gradient ofthe lineand so
= 0.1139
logb = 0.1139, thatisb= 10 0.1139 = 1.2999
The vertical intercept isloga. From the graph the vertical intercept islog4 so that
loga=log4,
thatisa=4
We conclude that the relationship betweenyandxisgiven byy = 4(1.3) x .