082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

80 Chapter 2 Engineering functions
2.4.3 Exponentialfunctions
Anexponentisanothernameforapowerorindex.Expressionsinvolvingexponentsare
called exponential expressions, for example 3 4 ,a b , andm n . In the exponential expressiona
x ,aiscalledthebase;xistheexponent.Exponentialexpressionscanbesimplified
and manipulated using the laws of indices. These laws aresummarized here.
a m a n =a m+n
a m
a n =am−n a 0 =1 a −m = 1
a m
(a m ) n =a mn
Example2.11 Simplify
a 3x a 2x
(a)
a 4x (b) a 2t (1 −a t ) +a 3t (c)
(a y ) 2
2a y
(d)
a −6z
a −2z
(e)
(2a 3r ) 2 a 2r
3a −5r
(f)
a x+y a y
a 2x
(g)
3a (x/y) a x
a y
a
Solution 3x a 2x
(a) = a5x
a 4x =ax
a4x (b) a 2t (1 −a t ) +a 3t =a 2t −a 3t +a 3t =a 2t
(c)
(d)
(e)
(f)
(g)
(a y ) 2
= a2y
2a y 2a = ay
y 2
a −6z
a −2z =a−6z−(−2z) =a −4z
(2a 3r ) 2 a 2r
= 4a6r a 2r
= 4a8r
3a −5r 3a −5r 3a
a x+y a y
=a x+2y−2x =a 2y−x
a 2x
3a (x/y) a x
= 3a (x/y)+x−y
a y
3
−5r
=
4a13r
Exponentialfunctions
Anexponential function, f (x), has the form
f(x)=a x
whereaisapositive constant called the base.
Some typical exponential functions are tabulated in Table 2.2 and are shown in Figure
2.26.Note from the graphsthatthese areone-to-onefunctions.
An exponential function is not a polynomial function. The powers of a polynomial
functionareconstants;thepowerofanexponentialfunction,thatistheexponent,isthe
variablex.