chap 1 index log

8/10/2019 Chap 1 Index Log

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Chapter 1 : Indices & Logarithms 1

CHAPTER 1 : INDICES &

LOGARITHMS

1.1 EXPONENT

1.2 LOGARITHMS

1.3 EXPONENT & LOGARITHMSEQUATION


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INTRODUCTION

Why study exponential & logarithmic

functions?


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They are very important in many technical areas, suchas 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, forexample population, compound interest & charge incapacitor.

We can also have exponentially decay for example

radioactive decay. Logarithm is a method of reducing long multiplications

into much simpler additions (and reducing divisionsinto subtractions).


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1.1 EXPONENT

Law Example Try

aman= am+n x3x7=x3+7=x10 x2x-5=

aman=am-n

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

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

264

6

4 kk

k

k

Law of exponents

2

5

h

h


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1.1 EXPONENT

Law Example Try

n

nn

b

a

b

a

n

nn

a

bba

Law of exponents

n

n

aa

1

8116

32

32 4

44

2

4w

8

1

2

12

3

3 23

425

2

552

2

22

3

34

Ex 2 pg 5


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Radical : the positive square root of

1.1 EXPONENT

abba nn means

Radical

a 0, b 0

nth root,nany +ve

integer

Ex 3 pg 6 - 7


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1.1 EXPONENT

Law Example Try

Properties of nth roots

0for

evenisif,

oddisif,

a

na

naa

n n

nnn baab

mnmn aa

n

n

n

b

a

b

a

3333 288 zzz 5 243m

3

2

27

8

27

83

3

3 4256

81

242564 3 729

612966

62166

44 4

33 3

8 8

5 5

6

2


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1.1 EXPONENT

Rational, - ve & Zero exponent

Rational exponent : n mm

nnm

aaa

10 a

m& n are

integers, n> 0

Negative exponent:n

n

a

a1

Zero exponent :

Ex 4 pg 89


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Logarithm function with base a, denoted by logais

defined by;

1.2 LOGARITHMS

813

8238log

12553125log

4

32

35

Definition

base

Exponent

(index / power)

formlexponentiaformlogaritmic

log yaxy xa

Example:

Equivalent

form


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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

yye

logln

Any

base Base

10

Base e

1.2 LOGARITHMS


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Example 1

1. Rewrite each function below in exponential orlogarithm form.

a) 72= 49

b) Log2128 = 7

c) 5-2= 1/25

d) Logb1=0

2. Determine the value of log27 and log312.

8074.22log

7log

7log 10

10

2

Ex 5 & 6 pg 1011


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Logarithms Example

logaxy= log

ax+ log

ay log

45x= log

45 + log

4x

loga(x/y) = logax logay ln 8ln 2 = ln (8/2) = ln 4

loga(xn) = nlogax log 5

3 = 3log 5

logaa = 1 log33 = 1

loga1 = 0 ln 1 = 0

Law of logarithms

1.2 LOGARITHMS


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Example 2

1. Use the property of logarithms to rewrite each of thefollowing:

a) ln 18 = ln (2.3.3) =

b) log 5 + log 2 =

c) log (3/5) =d) log 8x2log 2x=e) Log 1003.4= log (102)3.4=

2. Simplify & determine the value of ;2log 5 + 3log 44log 2

Ex 7 pg 1213


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Example 2

3. 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


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Exponential Equation The variable occurs

in the exponent.

E.g. 2x= 7

To solve:1) Use the properties of

exp.

2) Rewrite in equivalentform.

3) Solve the resultingequation.

Logarithmic Equation A logarithmof the

variable occurs.

E.g. log2 (x+2) = 5

To solve:1) Use the properties of

log.

2) Rewrite in equivalentform.

3) Solve the resultingequation.

1.3 EXPONENTIAL & LOGARITHS

EQUATION


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calculatorause,forSolve43ln

81ln

exponent)thedown(bring3Law81ln3ln

sideeachoflnTake81ln3ln

xx

x

x

Example 3

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

b) 52x+1= 254x-1

2

1

forSolve2812

4Lawand3LawlyApp5log285log12

sideeachoflogTake5log5log

55

55

528

512

5

14212

x

xxx

xx

xx

xx

Ex 10 pg 1516


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forSolve0.4582x

lnofProperty0.91632x

sideeachoflnTake5.2lnln

8byDivide

8

20

208

2

2

2

x

e

e

e

x

x

x

Example 3

Solve each of the following;c) 8e2x= 20

Ex 14 pg 1920


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Example 4

0127 24 kkSolve for k, if

aaaaa

ka

kk

forsolve&Factor043

factorizeandReplace0127

;Let

exponentofLaw0127

2

2

222

24

4

4

04

2

k

k

a

a

3

3

3

03

2

k

k

a

a

Step 1

Step 2

Step 3


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Example 5

aaa

aa

a x

xx

forsolve&Factor023

factorizeandReplace06

;5Let

exponentofLaw0655

2

2

25

2

02

x

a

a

Solve for mif 52x

- 5x

6 = 0Step 1

Step 2

Step 3

0.6826x

ln5

3lnx

3ln5ln

35

3

03

x

x

a

a Or

No solution,xisundefined

Ex 13 pg 8


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calculatorUse1096.6

formEquivalent7

ex

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

b) log2 (x+2) = 5

c) Log2(25x) = 3

30

forSolve232

formlExponentia22

52log

5

2

xx

x

x

Example 6


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2byDivide5000formlExponentia102

3byDivide42log

4Subtract122log3

4

xx

x

x

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

e) c

Example 6

4ln3ln2 xx

chap 1 index log

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