For UniKL Business School Student rare slide

1. CHAPTER 2 : EXPONENTS
& LOGARITHMS
2.1 EXPONENT
2.2 LOGARITHMS
2.3 EXPONENT & LOGARITHMS
EQUATION

2. INTRODUCTION
Why study exponential & logarithmic
functions?

3.  They are very important in many technical areas,
such as business, finance, nuclear technology,
acoustics, electronics & astronomy.
 Many of the applications will involve growth
(INCREASING) or decay (DECREASING).
 There are many things that grow exponentially, for
example population, compound interest &
charge in capacitor.
 We can also have exponentially decay for
example radioactive decay.
 Logarithm is a method of reducing long
multiplications into much simpler additions (and
reducing divisions into subtractions).

4. 2.1 EXPONENT
Definition
 If a is any real number & n is a positive
integer, then the n – th power of a is ;
Exponent
(index / power)
a n = a × a × .... × a
Example:
base
x
Graph the function f ( x ) = 2
Solution;
Produce the table values of x from -2 to 3.
x
-2
f(x)
0.25
-1
0
1
2
3

5. 2.1 EXPONENT
Law of exponents
Law
Example
am × an = am+n
x3x7 = x3+7 = x10
a ÷ a =a
m
n
m-n
k4
= k 4− 6 = k − 2
k6
(am)n = amn
(43)2 = 43(2)=46
(ab)n = anbn
(2b)3 = 23b3= 8b3
Try 
x2x-5 =
h5
=
−2
h
(55)2 =
(3xy)4 =

6. 2.1 EXPONENT
Law of exponents
Law
Example
n
2 4 16
2
  = 4 =
81
3
3
4
 a  an
  = n
 b b
1
a = n
a
−n
 a
 
 b
−n
2
n
b
= n
a
−3
2
 
5
1 1
= 3 =
8
2
−2
=
2
5
25
=
4
22
Try 
2
w 
  =
4
3 −2 =
4
 
3
−3
=

7. 2.1 EXPONENT
Radical, Rational, - ve & Zero exponent
 Radical : √  “ the positive square root of “
n – th root,
n any +ve
integer
n
a = b means b n = a
a ≥ 0, b ≥ 0
 Rational exponent : a
m
n
=
( a)
n
m
= n am
m & n are
integers, n > 0
 Zero exponent : a 0 = 1
 Negative exponent : a
−n
1
= n
a

8. PRACTICE 1
1. Evaluate the expression.
(
a) 4 −3 ⋅ 4 5
( )
b) 3 −2
c) 9( 9 )
a) 25 x 3 y 4
(
3
)(
)
−1 / 2
b) 4 x −2 − 3 x 5
−1 / 2
 1  2 
d)  −  
 3  


2. Simplify the expression.
−3
( − 3) 4 ( − 3) 5
e)
( − 3) 8
f) 23 / 4 ⋅ 4 −3 / 2
6a −4
c) −3
3a
)
d) y −3 / 2 y 5 / 3
(
e) 4 x 2 y 2 z 3
f)
)
2
( x + y )( x − y )
( x − y )0

9. 2.2 LOGARITHMS
Definition
 Logarithm function with base a, denoted by loga is
Exponent
defined by;
Equivalent
(index / power)
base
loga y = x ⇔
 
 
logaritmic form
x
a 
=y

exponential form
3
Example: log5 125 = 3 ⇔ 5 = 125
log2 8 = 3 ⇔ 23 = 8
3 4 = 81 ⇔
form

10. 2.2 LOGARITHMS
Type of Log
 Common logarithm : Logarithm with base 10, denoted
by,
log y = log y
10
 Natural logarithm : Logarithm with base e, denoted by
ln y = loge y
 Base conversion :
loga x log10 x ln x
logb x =
=
=
loga b log10 b ln b
Any
base
Base
10
Base e

11. Example 1
1. Rewrite each function below in exponential or
logarithm form.
a) 72 = 49
b) Log2128 = 7
c) 5-2 = 1/25
d) Logb1=0
2. Determine the value of log27 and log3 12.
log10 7
log2 7 =
= 2.8074
log10 2

12. 2.2 LOGARITHMS
Law of logarithms
Logarithms
loga xy = loga x + loga y
loga (x/y) = loga x − loga y
loga (xn) = n loga x
Example
log 45x = log 45 + log 4x
ln 8 – ln 2 = ln (8/2) = ln 4
log 53 = 3log 5
loga a = 1
log33 = 1
loga 1 = 0
ln 1 = 0

13. Example 2
1. Use the property of logarithms to rewrite each of the
following:
a) ln 18 = ln (2.3.3) =
b) log 5 + log 2 =
c) log (3/5) =
d) log 8x2 – log 2x =
e) Log 1003.4 = log (102)3.4 =
2. Simplify & determine the value of ;
2log 5 + 3log 4 – 4log 2

14. PRACTICE 2
If log 2 = 0.3010, log 3 = 0.4771 and log 5 = 0.6990,
determine each of the following without using a
calculator:
a) log 6 = log 2x3 = log 2 + log 3
= 0.3010 + 0.4771 = 0.7781
b) log 81
c)
log 1.5
d)
log √5
e)
log 50

15. 2.3 EXPONENTIAL & LOGARITHS
EQUATION
 Exponential Equation



The variable occurs in
the exponent.
E.g. 2x = 7
To solve:
1) Use the properties of
exp.
2) Rewrite in equivalent
form.
3) Solve the resulting
equation.
 Logarithmic Equation



A logarithm of the
variable occurs.
E.g. log2 (x+2) = 5
To solve:
1) Use the properties of
log.
2) Rewrite in equivalent
form.
3) Solve the resulting
equation.

16. Example 3
Solve each of the following;
a) 3x = 81
ln 3 x = ln 81
x ln 3 = ln 81
ln 81
x=
=4
ln 3
Take ln of each side
Law 3 (bring down the exponent)
Solve for x, use a calculator
b) 5 2 x +1 = (254x-1−1
52x+1 = 5 2 ) 4 x
log5 5 2 x +1 = log5 5 8 x −2
( 2x + 1) log5 5 = ( 8 x − 2) log5 5
2x + 1 = 8 x − 2
1
x=
2
Take log5 of each side
App ly Law 3 and Law 4
Solve for x

17. Example 3
Solve each of the following;
c) 8e2x = 20
8e 2 x = 20
20
8
= ln 2.5
e2x =
ln e 2 x
2x = 0.9163
x = 0.4582
Divide by 8
Take ln of each side
Property of ln
Solve for x

18. Example 4
Solve each of the following;
a) ln x = 7
x = e7
≈ 1096.6
Equivalent form
Use calculator
b) log2 (x+2) = 5
log2 ( x + 2) = 5
x + 2 = 25
x = 32 − 2
= 30
c) Log2(25 – x) = 3
Exponentia l form
Solve for x

19. Example 4
Solve each of the following;
d) 4 + 3log 2x = 16
3 log 2 x = 12
Subtract 4
log 2 x = 4
Divide by 3
2 x = 10 4
x = 5000
Exponentia l form
Divide by 2
2 ln 2 + ln x = ln 3
e) C
2 ln( x + 3 ) = ln( 4 x + 12)
f)
c

20. PRACTICE 3
Solve each of the following.
a) e 0.4t = 8
b) 5e −2t = 6
c) 12 − e 0.4t = 3
d) log2 x = 3
e) log3 27 = 2 x
f) log2 ( 2 x + 5 ) = 4
g) log2 x − log2 ( x − 2) = 3
h) log3 ( x + 1) − log3 ( x − 1) = 1