Understanding Engineering Mathematics

We need to say something about the definition ofaxfor different values ofx, building on
the work in Section 1.2.7. Ifx=nis a positive integer, then for any real numberawe
know that we can define

ax=an≡a︸×a×···×︷︷ a︸

nfactors

simply in terms of elementary algebraic operations.
Similarly, ifx=−nis a negative integer then we can define

ax=a−n=(a−^1 )n=a︸−^1 ×a−^1 ︷︷×···×a−^1 ︸

nfactors

And of course, by definition

a^0 = 1

In these definitions there is no need for any qualification on the real numbera– it can be
positive, negative, rational, irrational.
We can extend the definition ofaxto the case whenxis a fraction or rational number

by defininga

1

q,whereqis an integer, as the positive real number such that when raised

to the powerqit yieldsa:

(a

1

q)q≡a

We s a ya

1

q is theqth rootofa, sometimes writtenq√a. However, complications arise

ifais negative, so henceforth we insist thatais positive, thenaxwill always be a real
function.
Forx=p/qa rational number we can now defineaxby

ax=a

p

q=(a

1

q)p

So, providedxis a rational number, anda>0, theexponential function,ax,isawell
defined function ofx. The extension to the case whenxis anirrational numberis not a
trivial step, but we will assume that it can be done and therefore thataxis in fact defined
for all real values ofx, rational or irrational, providedais positive.

Solution to review question 4.1.2

A.Iff(x)=axthenf(y)=ayand so

f(x)f(y)=axay=ax+y=f(x+y)

f(x−y)=ax−y=ax/ay=f(x)/f(y)

B. We simply apply the rules of indices but with the indices as functions

ofx.

(i)axa−x=ax−x=a^0 = 1