CHAPTER 2 : EXPONENTS & LOGARITHMS

2.1 EXPONENT2.2 LOGARITHMS2.3 EXPONENT & LOGARITHMS EQUATION

INTRODUCTION

Why study exponential & logarithmic functions?

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).

xxf 2)( Example:Graph the function Solution;Produce the table values of x from -2 to 3.

If a is any real number & n is a positive integer, then the n – th power of a is ;

2.1 EXPONENT

aaaan ....

Definition

x -2 -1 0 1 2 3

f(x) 0.25

base

Exponent (index / power)

2.1 EXPONENT

Law Example Try

am × an = am+n x3x7 = x3+7 = x10 x2x-5 =

am ÷ an =am-n

(am)n = amn (43)2 = 43(2)=46 (55)2 =

(ab)n = anbn (2b)3 = 23b3= 8b3 (3xy)4 =

Law of exponents

2646

4 kk

k

k 2

5

h

h

2.1 EXPONENT

Law Example Try

Law of exponents

n

nn

b

a

b

a

n

nn

a

b

b

a

nn

aa

1

81

16

3

2

3

24

44

2

4

w

8

1

2

12

33 23

4

25

2

5

5

22

22

3

3

4

2.1 EXPONENT

Radical, Rational, - ve & Zero exponent Radical : √ “ the positive square root of “

abba nn means

a ≥ 0, b ≥ 0

n – th root, n any +ve

integer

Rational exponent : n mmnnm

aaa

10 a

m & n are integers, n > 0

Negative exponent : nn

aa

1

Zero exponent :

PRACTICE 1

1. Evaluate the expression.

2/34/3 42 f)

53 44 a)

2/199 c)

323 b)

32

3

1 d)

2/14325 a)

yx

8

54

3

33 e)

52 34 b) xx

3

4

3

6 c)

a

a

3/52/3 d) yy

23224 e) zyx

0

f)yx

yxyx

2. Simplify the expression.

Logarithm function with base a, denoted by loga is defined by;

2.2 LOGARITHMS

813

8238log

12553125log

4

32

35

Definition

base

Exponent (index / power)

form lexponentiaform logaritmic

log yaxy xa

Example:

Equivalent form

b

x

b

x

b

xx

a

ab ln

ln

log

log

log

loglog

10

10

Common logarithm : Logarithm with base 10, denoted by,

Natural logarithm : Logarithm with base e, denoted by

Base conversion :

Type of Log

yy 10loglog

yy elogln

Any base Base

10

Base e

2.2 LOGARITHMS

Example 1

1. Rewrite each function below in exponential or logarithm form.

a) 72 = 49 b) Log2128 = 7c) 5-2 = 1/25d) Logb1=0

2. Determine the value of log27 and log3 12.8074.2

2log

7log7log

10

102

Logarithms Example

loga xy = loga x + loga y log 45x = log 45 + log 4x

loga (x/y) = loga x − loga y ln 8 – ln 2 = ln (8/2) = ln 4

loga (xn) = n loga x log 53 = 3log 5

loga a = 1 log33 = 1

loga 1 = 0 ln 1 = 0

Law of logarithms

2.2 LOGARITHMS

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

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 = 0.7781b) log 81

c) log 1.5

d) log √5

e) log 50

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.

2.3 EXPONENTIAL & LOGARITHS EQUATION

Example 3

calculator a use, for Solve 43ln

81ln

exponent) the down (bring 3Law 81ln3ln

side each of ln Take 81ln3ln

xx

x

x

Solve each of the following;a) 3x = 81

b) 52x+1 = 254x-1

2

1

for Solve 2812

4Law and 3Law ly App5log285log12

side each of log Take 5log5log

55

55

528

512

5

14212

x

xxx

xx

xx

xx

for Solve 0.4582x

ln ofProperty 0.91632x

side each of ln Take5.2lnln

8by Divide8

20

208

2

2

2

x

e

e

e

x

x

x

Example 3 Solve each of the following;

c) 8e2x = 20

calculator Use 1096.6

form Equivalent 7

ex

Solve each of the following;a) ln x = 7

b) log2 (x+2) = 5

c) Log2(25 – x) = 3

30

for Solve 232

form lExponentia 22

52log5

2

xx

x

x

Example 4

Example 4

2by Divide5000

form lExponentia102

3by Divide42log

4 Subtract122log3

4

x

x

x

x

Solve each of the following;d) 4 + 3log 2x = 16

e) C

f) c

3lnln2ln2 x

124ln3ln2 xx

PRACTICE 3

Solve each of the following.

452log f) 2 x

8 a) 4.0 te

312 c) 4.0 te

65b) 2 te

3log d) 2 x

32loglog g) 22 xx

x227log e) 3

11log1log h) 33 xx