INDICES, SURDS AND LOGARITHM - Wikispaceslydiavalensia.wikispaces.com/file/view/math_book+surd,+indices+and... · Mathema tics is the way you think INDICES, SURDS AND LOGARITHM

1

Lydia Valensia

X Grade

Mathematics is the way you think

INDICES, SURDS AND

LOGARITHM

www.freedomroad.org.uk

2

GRADE X

SMA Negeri 2 Sekayu ___Lydia Valensia Mathematics is the way you think

Hello, my students….

Today, we are going to learn about indices, surds and logarithms.

You know, actually indices, surds and logarithms are closely related.They

are most of the time,studied together. So now, lets begin…

STANDARD COMPETENCE:

1. Solving problems related to indices, surds and logarithms BASIC COMPETENCE:

Using laws of indices, surds and logarithms.

Doing the algebraic manipulation in computation related to indices, surds and logarithms.

(Integrated with E- SETS )

3

GRADE X


1.1 INDICES

A. PROPERTIES OF EXPONENT

Have you ever bought eggs ? How many eggs that you

get, if you buy 8 eggs each day for 8 days?

Ok. This is easy question, isn’t it?

So, what is the result? Yups, that is right 64.

Do the result 64 comes from 8 x 8 ?. We can write 8 x 8 = 82.

It means that we have used indices.

What is indices ?

Now, Lets discuss it.

Indices or powers are also called exponent

The exponent of a number says how

many times to use the number in

multiplication.

In example 82 = 8 x 8 = 64. How to

read 82? Don’t miss it !

Need more example? Here they are.

35 = 3 x 3 x 3 x 3 x 3

In words : 35could be called “ 3 to the fifth power”, “3 to the power 5”.

And exponents make it easier to write and use many multiplications.

Example :

Recurring Number Magic

Activity: You write down the following 8 digit number on a piece of paper:

1 2 3 4 5 6 7 9

Then ask a friend to circle one of the digits. Say that they circle number 7.

You then ask your friend to multiply the 8 digit number by 63, and magically the result ends up being:

1 2 3 4 5 6 7 9 x 6 3 7 7 7 7 7 7 7 7 7

with the answer as a row of the chosen number 7.

How about if your friends circle number 3? Ask them to multiply by 27 and the result is 333333333.

What is the secret ?

In words: 82 could

be called ―8 to the second power‖, ―8 to the power 2‖ or simply ―8 squared‖.

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GRADE X


116 is easier to write and read than 11 x 11 x 11x 11 x 11x 11

You can multiply any number by itself as many times as you want using this

notation.

So, in general:

THE KEY OF EXPONENT

The "Laws of Exponents" (also called "Rules of Exponents"), all come from

three following ideas:

1. The exponent of a number says to multiply the number by

itself so many times

2. The opposite of multiplying is dividing, so a negative

exponent means divide

3. A fractional exponent like 1/n means to

take the nth root:

If you understand those, trust me that you are able to continue

next journey of exponent.

All the laws below are based on those ideas.

Laws of Exponents

Here are the Laws (explanations follow):

Law Example

x1 = x 61 = 6

x0 = 1 70 = 1

x-1 = 1/x 4-1 = ¼

xmxn = xm+n x2x3 = x2+3 = x5

xm/xn = xm-n x4/x2 = x4-2 = x2

(xm)n = xmn (x2)3 = x2×3 = x6

(xy)n = xnyn (xy)3 = x3y3

(x/y)n = xn/yn (x/y)2 = x2 / y2

x-n = 1/xn x-3 = 1/x3

Puzzle Math Aptitude Test Which of the following sentences is correct? Nine and five are thirteen. or Nine and five is thirteen. Solution: Neither is correct ,

9 + 5 = 14.

http://www.mathsisfun.com/algebra/exponent-fractional.html

5

GRADE X


Laws Explained

The first three laws above (x1 = x, x0 = 1 and x-1 = 1/x) are just part

of the natural sequence of exponents.

Have a look at this example:

Simplify and write down in positive exponents

1. 4

422

2

2

2

75 1.

yyy

y

y

y

yy

3

31

)3(6)3(2

3362

313232

323

2

1

2

.2

2.2

2.22

.2.2

a

a

a

aa

aaa

a

3. Evaluate the following without using calculator :

(a) 33

4

88

25

9

5

3

5

3

5

3

125

27

125

272

2

3

32

3

32

33

2

b

If you find it hard to remember all of these rules, then remember this:

“ You can always work them out if you understand the three that

have explained”.

Ups, what will happen if x (variable) = 0

Positive Exponent (n>0) 0n = 0

Negative Exponent

(n<0) Undefined! (Because dividing by 0)

Exponent = 0 Ummm ... (see column did you

know)

Did You Know

The Strange case

of 00

There are two

different arguments

for the correct

value. 00 (zero to the

power zero) could be

1, or possibly 0, so

some people say it is

really “

indeterminate” X0 = 1, so 00 = 1

00 = 0, so 00 = 0

When in doubt

00 = “

indeterminate”

The example use “y and a” as

variables not x. It can be

changed to another alphabet

The question can be

solved using the formula

6

GRADE X


EXERCISE 1 “ Cogito Ergo Sum--Descrates”

1. Simlplify:

(a). 4

33

4.2

(d). 3a3 x 2ab2

(b). 62

33

3.)7(

9.)7(

(e). (2x3)-3

(c). r3 x r (d). s6 / s2

2. Simplify the expressions.

(a). ( ) ( ) ( )

( ) ( ) (b).

3. If x = 38, express x½ in the form 3n where n is an integer.

4. Given a = 29 x 5-6 Express a1/3 and a-1 in the form 2m x 5n where m and

n are integers.

5. The number 450 can be written as 2a x 3b x 5c.

Calculate the values of a, b, and c

B. EXPONENTIAL EQUATIONS

An exponential equation is an equation that contains a

variable with an exponent, or a variable in an exponent. For

example 5x = 5

3, 10

1–x = 10

4.

To solve exponential equations, you need to have equations

with comparable exponential expressions on either side of the

“equals” sign, so you can compare the power.

Let * + (

) If then m = n.

Did You Know Real Number

R

{

}

Q Rational Number

{

}

Z Integer

* +

Whole Numbers

Natural

Numbers

* +

* +

{ √ }

Irational

Number

7

GRADE X


Rule

If then {

Take a look at the example.

Solve

1. 32x–1 = 27 2. = 8.

32x–1 = 27 24x2+4x

= 23

32x–1 = 33 4x2 + 4x = 3

2x – 1 = 3 4x2 + 4x – 3 = 0

2x = 4 (2x – 1)(2x + 3) = 0

x = 2 x = 1/2 ,

–3/2

EXERCISE 2 “ Mathematics is universal language”

Find the solution of the exponential equations below

1. (a). If then x = … (b).If ( ) then x = ……

2.

3.

4. ( )

5.

6.

7.

8. Solve the equation

9. The formulae for the volume and the surface area of a sphere are V =

and respectively, where r is the sphere’s radius. Find

expressions for

(a). S in terms of V (b). V in terms of S

8

GRADE X


10. The moving kinetic energy, K joules, possessed by an object of mass

m kg moving with speed v ms-1 is given by the formula K =

Find

the kinetic energy possessed by a bullet of mass 1.5 x 10-3 kg moving

with speed 12 x 103 ms-1.

1.2 SURDS

Well , having been exposed to indices in the previous lessons will help you

understand the use of surds. In this section, we use the word “surd”

means not only a number with radical sign. Numbers whose square roots

cannot be determined in terms of rational numbers eg. √ etc are called

surds. Such numbers occur frequently in Trigonometry when calculating

the ratio of angles eg. Cos 30= √

, tan 60 = √ and in coordinate geometry

in the calculation of distances. Therefore You will find it useful to have a

sound knowledge of surds. Now lets begin our journey in SURDS.

A. UNDERSTANDING SURDS

What is the exactly result of 43 and

You get the result and 24

You can’t expressed exactly 3 (it is a surd) but 4 can be simplified (it

equals 2). 4 isn’t a surd.

Can you define the surd

Sometimes it is useful to work in surds, rather than using an approximate

decimal value. Surds can be manipulated just like algebraic expressions

and sometimes it may be possible to eliminate the surd (called

rationalizing the expression), which may have not been possible if you

tried to work with approximate value. When asked to give the

approximate decimal answer will not do and you will have to manipulate

surds in order to give final answer in simplified surd form.

Need more example ?

Have a look at these:

Number Simplifed As a Decimal Surd or

not?

√ √ 1.4142135(etc) Surd

√ √ 1.7320508(etc) Surd

√ 2 2 Not a surd

A Surd is an expression containing a root with an irrational solution that can

not be expressed exactly

History

How did we get the

word "Surd" ?

Well

around 820 AD al-

Khwarizmi (the

Persian guy who we

get the name

"Algorithm" from)

called irrational

numbers "'inaudible"

... this was later

translated to the Latin

surdus ("deaf" or

"mute")

9

GRADE X


√

0.5 Not a surd

√

√

2.2239800(etc) Surd

√

2 2 Not a surd

√

√

1.2457309(etc) Surd

As you can see, the surds have a decimal which goes on forever without

repeating, and that makes them irrational number. It is called irrational

because it cannot be written as a ratio (or fraction), not because it is

crazy . But if it is a number, it can be written as a simple fraction then

it is called a rational number.

The conclusion:

Laws of Surds

If it is a root and irrational, it is a surd.

But not all roots are surds.

History

Irrational Numbers

Apparently Hippasus (one of Pythagoras' students) discovered irrational numbers when trying to represent the square root of 2 as a fraction (using geometry, it is thought). Instead he proved you couldn't write the square root of 2 as a fraction and it was irrational.

However Pythagoras could not accept the existence of irrational numbers, because he believed that all numbers had perfect values. But he could not disprove Hippasus' "irrational numbers" and so Hippasus was thrown overboard and drowned! Oh…. Dear!

However

Pythagoras

could not accept the

existence of irrational

numbers, because he

believed that all numbers

had perfect values. But he

could not disprove

If a, b, (a ≠ 0 and b ≠ 0 are positive) and (m and n are real numbers)

then

Multiplication of Surds :

1. abbxabxabxa 2

12

1

2

1

Division of Surds

b

a

b

a

b

a

b

aba

2

1

2

1

2

1

:.2

Addition of Surds:

anmanam .3 4. √ √ = √ √ (where a = m + n and b = m. n)

√ √ = √ - √ ( )

Substraction of Surds:

anmanam .5

Applying

mmm bxabxa

Applying m

m

m

b

a

b

a

10

GRADE X


To make you easy using surds, follow these steps:

1. Check which numbers are of the same base and which are of the

same power.

2. Use appropriate formulate.

Note : a means the positive square root of a, while - a means

the negative square root of a.

Have a look at the following examples:

1. Evaluate the following surds without the use of a calculator

(a). 182 x (b) 6

5

3

40

Solution:

(a). 636)8)(2(182 x (ans)

(b). 4165

6

3

40

6

53

40

6

5

3

40

(ans)

2. Evaluate the following surds, without the use of a calculator.

(a) 33

23

2

1 (b) 5354

Solution:

(a) 36

73)

3

2

2

1(3

3

23

2

1 (ans)

(b). 55345354 (ans)

The question can be solved using the formula

b

aba


abbxa


anmanam


anmanam

√

√

√

√

√

√

√

11

GRADE X


EXERCISE 3

“ Plan for tomorrow but live for today”

1. Write as simply as possible

(a). 40290

(b). 402903

(c). 40390 x

2. Simplify the expression

(a). √ √ √

(b). √ √ √

(c). √

√

√

3. ABCD is a rectangle in which AB = √ cnd BC = √ . Giving each

answer in simplified form, find

(a). The area of the rectangle (b). The length of the diagonal AC

4. Find the length of the third side in each of the following right-

angled triangles, giving each answer in simplified surd form.

(a). √ (b). √ cm

cm 7 cm

5. Simplify the following questions

(a). √ √

(b). √ √ √ √

(c). √ √ √ √

√

6. Let a, b, m and n be four real numbers, satisfying a = m + n and

b = m. n. Then, proof that

(a). √ √ √ √

(b). √ √ √ √ , (m > n)

Did You Know

√ and

have different

meanings in the set

of all real numbers.

√ √ = 4

If then

or

√

Isaac Newton (1642-1727)

Isaac Newton is thought by many to have been one of the greatest intellects of all time. He went to Trinity College Cambridge in 1661 and by the age of 23 he had made three major discoveries: the nature of colours, the calculus and the law of gravitation. He used his version of the calculus to give the first satisfactory explanation of the motion of the Sun, the Moon and the stars. Because he was extremely sensitive to criticism, Newton was always very secretive, but he was always very secretive, but he was eventually persuaded to publish his discoveries in 1687.

12

GRADE X


7. (a). Find x, so that √ √ √ √

(b) Evaluate √ √ √ √

B. RATIONALISATION OF DENOMINATORS

Another technique to simplify expressions involving surds is to rationalise

the denominator. This means removing a surd from a fraction.

You can rationalize the denominator easily, just follow two kinds of

method below. Lets check it out !

Well, you have already known the steps. Practice will makes you

become perfect so lets look the examples before doing the

exercise

1. Rationalise the denominator of the following surds:

(a). 11

2 (b)

25

3

(c).

625

1

Solution:

(a). 1111

9

11

11

11

2

11

2

(b).

45

)25(3

25

253

25

25

25

3

25

3

22

1. Denominator has a single-term surd

If the denominator has a surd of the form a

Step : Rationalise it by multiplying the numerator and the denominator by

a to get

2. Denominator has sum or difference of surds

If the denominator is of the form anam

Step : Rationalise it bymultiplying the numerator and the denominator by its

conjugate surd anam to get a rational number m2a – n2b.

Puzzle

Math Aptitude Test

1. Take two apples from three apples and what do you have?

2. Some months have 30 days, some have 31; how many have 28 days?

Solution:

1. You have two apple

2. All of the months

have 28 days.

13

GRADE X


(c). 2425

625

625625

625

625

625.

625

1

EXERCISE 4 “ What oxygen is to lungs, such is hope to meaning life”

1. Rationalize the denominator in each of the following expression and

simplify them

a.

√ b.

√

√

2. Rationalize the denominator and simplify these fractions.

a.

√ c.

√

√ √

b. 643

1

d.

37

3

3. Evaluate √ √ √ √

√

4. a. Explain why

√

√

√

√ and hence show that

√

√

b.Show that

√ √

√ √

5. Find x if √√

√

14

GRADE X


1.3 LOGARITHM

Hmmm, hello… ..How are you doing?

In the previous lesson, you have learned about indices, aren’t you?. How is

it going? All of them seem easy for you? Can you solve this . Is it

a kind of indices? The solution remain a mystery, don’t it?

Well. What we are going to learn now is answering that mystery.

Logarithm answer your curiosity related to indices. I believe that all of

you ever think this kind of question.

In its simplest form, a logarithm answers a simple question above

Example

The question is an easy question and every one have the same idea to

answer that. But, how to write it?

We would write "the number of 2s you need to multiply to get 8 is 3" as

log2(8) = 2log(8) = 3

So, it means these two things are same:

Base

The number we are multiplying is called the "base", so we would say:

"the logarithm of 8 with base 2 is 3"

or "log base 2 of 8 is 3"

or "the base-2 log of 8 is 3"

So a logarithm also answers the question

What exponent do we need

(For one number to become another number)?

For the next,

we will write alogb , where

“a” is the base

How many of one number do we multiply to get another number?

How many 2 need to be multiplied to get 8?

Answer: 2 x 2 x 2 = 8, so we needed to multiply 3 of the 2 to get 8

15

GRADE X


Have a look at this example:

1.What is 10log (100) ……?

Ok. To answer the this question, you can back to our first question at

the first session in logarithm part. “What exponent do we need

(For 10 to become 100)?”

Answer:

102 = 100, so an exponent of 2 is needed to make 10 into 100.

It means that, 10log (100) = 2

2. What is 3log (81)…?

Answer:

34 = 81, so an exponent of 4 is needed to make into 8.

3log 81 = 4

3. If 7,05log x show that 7 355x

Answer:

)(55

55

5

5

57,05log

7 3

7

3

7

107

10

10

7

10

7

7,0

qedx

xx

xx

x

xx

So, After looking the apperception above

a logarithm is the exponent to which the base must be raised to produce

a given number

Based on the previous explanation, we can define a number as

DEFINITION

History

― …il cessa de calculer et de vivre — … he ceased to calculate and to live ‖.

Leonhard Paul

Euler April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. Euler is considered to be the preeminent mathematician of the 18th century and one of the greatest of all time. He is also one of the most prolific; his collected works fill 60–80 quarto volumes.[3] A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all. A Logarithm is the exponent to which the base must be raised to produce

a given number

agifonlyandifxa xg log

http://en.wikipedia.org/wiki/April_15

http://en.wikipedia.org/wiki/1707

http://en.wikipedia.org/wiki/September_18

http://en.wikipedia.org/wiki/Old_Style_and_New_Style_dates

http://en.wikipedia.org/wiki/1783

http://en.wikipedia.org/wiki/Swiss_%28people%29

http://en.wikipedia.org/wiki/Mathematician

http://en.wikipedia.org/wiki/Physicist

http://en.wikipedia.org/wiki/Russia

http://en.wikipedia.org/wiki/Germany

http://en.wikipedia.org/wiki/Bookbinding#Terms_and_techniques

http://en.wikipedia.org/wiki/Leonhard_Euler#cite_note-volumes

http://en.wikipedia.org/wiki/Pierre-Simon_Laplace

16

GRADE X


EXERCISE 5 “When life give you 100 reasons to cry, then show life that you have 1000 reasons to smile”.

1. Write in exponential form:

a). 664log2 d). 6125log5

b). 327

1log3 e). 23log3

c). 38

1log2

1

f). 201,0log10

2. Find the value of the following.

a). 32log2 f). 121

1log11

b). 49log7 g). 1000log

c). 1log3 h). 25log5

d). 243

1log3 i). 3log4

4

e). 11log13 j). 4log3

3

1

3. a). If 4,02log x show that 28

1x .

b). if 2

122log2

1

x , show that 24

3x

Common Logarithms : Base 10

Sometimes you will see a logarithm written without a base, like this :

Log(100). This usually means that the base is really 10

Common Logarithms : Base “e”

It is called a "common logarithm". Engineers love to use

it.

On a calculator it is the "log" button.

17

GRADE X


Another base that is often used is e (eulers number) which is

approximately 2, 71828.

This is called a "natural logarithm". Mathematicians use

this one a lot.

On a calculator it is the "ln" button.

It is how many times you need to use "e" in a multiplication, to get the

desired number.

Example: 389,771828,22389,7log389,7ln 2 becausee

You have learned various rules for manipulating and simplifying

expressions with exponents, such as the rule that says that x3 · x5 equals

x8 because you can add the exponents. There are similar rules for

logarithms.

Note: g is the base of logarithm which is satisfied 0 < g < 1 or g >1

(g > 0 and

g 1).

Before continuing the next journey about logarithm, it will be better for

you to see where do the rules of logarithm above come from.

Log Rules:

ag

aaiii

an

maii

bbai

gaii

g

aai

axna

bab

a

babxa

a

gng

gmg

gag

a

g

p

pg

gng

ggg

ggg

g

n

n

log.6

loglog).

loglog).

loglog.log)..5

log

1log).

log

loglog)..4

loglog.3

logloglog.2

loglog)log(.1

18

GRADE X


Can you imagine , if you do not know where do you come from, who is

your parents, where is your home town. You will feel that there is

something less in your life. It same like knowing the mathematics formula

without knowing how could be like that.

So, Lets notice following evidence

Rule 1 :

Prove: Suppose xag log maka xga ………..(1)

ybg log maka ygb ………..(2)

Multiply equation(1) to (2), we get:

yx gxgbxa

yxgbxa

yxgg gbxa log)(log

yxbxag )(log …………def logarithm

babxa ggg loglog)(log ………….(qed)

Rule 2 :

Prove : Divide equation 1 by equation 2, we get:

y

x

g

g

b

a

yxgb

a

yxb

ag

log

bab

a ggg loglog)(log ………….(qed)

Rule 3 :

Prove : axaxxaxaxaa gng ...loglog

n factor for each a

= aaaa gggg loglog...loglog

n times

babxa ggg loglog)log(

bab

a ggg loglog)log(

axna gng loglog

19

GRADE X


= n x ag log …………….(qed)

Ok. You have read some of the evidence of logarithm’s rules. You should

know all of them not only part of them,, so we give what you need to

improve your knowledge. Lets go!

Rule 4 :

Prove :

g

ax

gxa

ga

gathenxaSuppose

p

p

pp

xpp

xg

log

log

loglog

loglog

,log

).....(..........log

loglog qed

g

aa

p

pg

Substitute p = a at the right side, we get:

ga

g

aa

a

g

a

ag

log

1log

log

loglog

Rule 5:

Prove : i) a

bx

g

abxa ag

log

log

log

logloglog

ga

becomeequationtheapIf

b

aa

a

g

p

pg

log

1log

:,

log

loglog

The Rule 5 are the expand of the

previous rules

aaiii

an

maii

bbxai

gng

gmg

gag

n

n

loglog).

loglog).

logloglog).

20

GRADE X


g

b

log

log

bg log …………… (qed)

ii). n

mmg

g

aa

n

log

loglog

an

m

g

a

n

m

g log

log

log

……………. (qed)

iii) We leave the prove for you.

Do not think that you can’t ! Just do it. We believe you.

Rule 6 :

Prove : Suppose agthenxa xg ,log . I want you remember the

definition of logarithm.

So, thenxag ,log

ag

gg

a

xa

g

g

log

log

……………. (qed)

You should expect to need to know these rules, because there is a certain

type of question that the teacher can put on the test to make sure you

know how to use the rules; you won't be able to "cheat" with your

calculator. Here's what they look like:

Let blog(2) = 0.3869, blog(3) = 0.6131, and blog(5) = 0.8982. Using

these values, evaluate blog(10).

Since 10 = 2 · 5, then:

blog(10) = blog(2 · 5) = blog(2) + blog(5)

Since I have the values for blog(2) and blog(5), I can evaluate:

blog(2) + blog(5) = 0.3869 + 0.8982 = 1.2851

Then blog(10) = 1.2851.

Let blog(2) = 0.3869,blog(3) = 0.6131, and blog(5) = 0.8982. Using

these values, evaluate blog(7.5).

Rule 6 is the expand of logarithm

definition

ag ag

log

21

GRADE X


This one is a bit more complicated. But if you take a little time,

you will notice that 7.5 = 15 ÷ 2, so:

blog(7.5) = blog(15 ÷ 2) = blog(15) – blog(2)

And 15 = 5 · 3, so: Copyright © Elizabeth Stapel 2000-2007 All

Rights Reserved

blog(15) – blog(2)

= [blog(5) + blog(3)] – blog(2)

= blog(5) + blog(3) – blog(2)

And now I can evaluate:

blog(5) + blog(3) – blog(2)

= 0.8982 + 0.6131 – 0.3869

= 1.1244

Then blog(7.5) = 1.1244.

EXERCISE 6 “Never give up, never give in, and don’t let your weakness

win”

1. Find the value of :

(a). 6log18log 33 (f). 6log23log24log 777

(b). 4

11log5log 22 (g). 9log1 2

1

(c). 4log3320log 55 (h). 21log

81log

(d).

2

12log

2log5log 22 (i). 13log254log

(e) 5,2log4,0 (j). 4,2log2log6log 555

2. Simplify

(a) aa gg loglog 4 (c). )1log()1log( 2 aa gg

(b). 2

1log

1loglog

xxx (d) 23 logloglog xxx xxx

3. If x = 0,6666… and b = 0,4444….

22

GRADE X


Find the value of 2log

2

yx.

4. If x

xxf

log.21

log)(

3

3

, find f(x) + f ...

3

x

5. If log 2 = a and l og 3 = b, find:

(a). log 18 (b) 5log 6

6. Evaluate the value of :

8log.54log.72log.3

512log.8

7. Find the value of a, if it is known:

6

5

4

1log16log8log 3 aaa

8. Find the value of x which is satisfied equation:

(a). 6log)1(log)2(log xx

(b). 1)1(log)1(log 33 xx

9. Given that : log 2 = 0,301 , log 3 = 0,477 dan log 7 = 0,845. Evaluate :

(a). log 5 (c) log 6

(b) 3 5,10log (d) log 7

6

10. Using properties of logs, show that

1

4

1ln4ln

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