Understanding Engineering Mathematics

• logxα=αlogx
  Putx=as,sos=logx,thenxα=(as)α=asα=aαsand soαs=
  logxα=αlogx

The last result holds for any real numberα, positive or negative, rational or irrational.
Sometimes we need to change the base of a logarithm. Thus, suppose we have logax
and we wish to convert this to a form involving logbx,b=a,wehave

y=logax

so x=ay

}

( 4. 2 )

and therefore

logbx=logbay=ylogba

=logaxlogba

So

logax=

logbx

logba

In particular, ifx=bthis gives

logab=

1

logba

For completeness we will anticipate Chapter 8 here and mention that the derivative of the
natural log function is simply the reciprocal:

d

dx

(lnx)=

1

x

Equations of the form

ax=b

occur frequently in engineering applications and may be solved by using logs. Thus, taking
logs to baseawe have

loga(ax)=logab

=xlogaa=x

So
x=logab

The graph of the logarithm function follows easily from that of the exponential function,
its inverse, by reflecting the latter in the liney=x(101
➤
). This is shown in Figure 4.4.