Understanding Engineering Mathematics

(やまだぃちぅ) #1

Logs can also be used to simplify graphical representation of certain functions. Thus,
given any function of the form


y=kxα

We can take logs to any base and obtain


logy=log(kxα)
=logk+αlogx

If we now put


X=logxY=logy

then we obtain the equation:


Y=αX+logk

IfYis plotted againstXon rectangular Cartesian axes, as in Section 3.2.2, then this is
a straight line. As we will see in Section 7.2.4 this line has gradient, or slope,αand an
intercept on theyaxis of logk.


Solution to review question 4.1.7

A.If 2x+^1 =5 then taking natural logs of both sides gives

ln( 2 x+^1 )=(x+ 1 )ln 2=ln 5

So
x+ 1 =

ln 5
ln 2

= 2. 322

to three decimal places, sox= 1. 322
B. Ify= 3 x^6
then taking logs to any convenient baseawe have

logay=loga( 3 x^6 )
=loga(x^6 )+loga 3
=6logax+loga 3

Put
X=logaxY=logay

to get the form of a straight line equation:

Y= 6 X+loga 3

The gradient of this line is 6 and its intercept on they-axis is loga 3
(➤212).
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