Understanding Engineering Mathematics

4.3.2 The general exponential functionax

(i) 1 (ii) 12 (iii) 1

(iv) −

1

x

√

x^2 + 1

(v) e−^2 x(x+^1 ) (vi) a^2

(vii) a−^3

4.3.3 The natural exponential functionex

A. (i) £C

(

1 +

I

52560000

) 525600

(ii) £C

(

1 +

I

3153600000

) 31536000

B. (i) 2.7183 (ii) 1.1052 (iii) 7.3891

C.

x

y

x

y = ex − 1 y

y = 1 − e−x

1

0

D.x=0.

4.3.4 Manipulation of the exponential function

(i) eA+^2 B+^3 C (ii) e^2 (^2 x+y) (iii) ex

(^2) + 2 x− 2 y− 3
(iv) e−A+^4 B−^4 (v) ( 1 +e−^2 A)^2 (vi) e−A−^2 B+e−^3 B+eB−^2 A+e−A
4.3.5 Logarithms to general base
A.(i) 256 (ii) 8 (iii) e^4 (iv) 729
(v) 81 (vi) e^2
B.(i) 3 (ii) 4 (iii) 3 (iv) 2
(v)^12 (vi) 4 (vii) 7 (viii) 5
4.3.6 Manipulation of logarithms
A. (i) lne^17 = 17 (ii) log 2 (x^5 ) (iii) loga( 2 xy) (iv) 0
(v) ln(x(^3 +^2 /lna)) (vi) loga
(
x
1 −log^1
a^2 a
)
(vii) ln(x^1 +
4
lna)