This document introduces exponents, logarithms, and exponential and logarithmic equations. It defines exponents and logarithms, including common, natural, and base conversion. Laws of exponents and logarithms are presented along with properties of radicals, rational exponents, negative exponents, and zero exponents. Examples of solving exponential and logarithmic equations are provided. The summary provides an overview of the key topics and concepts covered in the document.

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