Understanding Engineering Mathematics

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).