Understanding Engineering Mathematics

The two results

x=elnxandx=lnex

expressing the fact that the exponential and the log are inverse functions of each other are
extremely important in advanced mathematics and are used repeatedly in, for example, the
solution of differential equations (Chapter 15).

Solution to review question 4.1.5

It is perhaps easiest to think of logaxas the power to whichamust be

raised to givex– then if we can expressxin the formay,ywill be the

valuewewant.

(i) log 101 =log 10100 = 0

In fact, log of 1 to any base (except 1!) is zero

(ii) log 22 =log 221 = 1

(iii) log 327 =log 333 = 3

(iv) log 2

( 1

4

)

=log 2 ( 2 −^2 )=− 2

Note that we can have negative logs – we just can’t take the log of

a negative number and expect a real number.

(v) loga(a^4 )= 4

(vi) loga(ax)=x

(vii) log 31 =log 330 = 0

(viii) lne=logee= 1

Of course,eis no different to any other log base in this respect.

(ix) ln

√

e=lne

1

(^2) =^12
(x) lne^2 = 2
(xi) ln 1= 0
4.2.6 Manipulation of logarithms
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120 137➤
Since the log is basically just an index or power, the properties of logs can be deduced
from the laws of indices. We find, for any basea(positive and not equal 1):

• log 1=0 becausea^0 = 1


• log(xy)=logx+logy
  Putx=as,y=atsos=logxandt=logythenxy=as+tsos+t=
  logx+logy=log(xy)


• log(x/y)=logx−logy
  Putx=as,y=atsos=logxandt=logythenx/y=as−tsos−
  t=logx−logy=log(x/y)