Unit1 Exp,Log,Surd MATH2(D) Ikbn (Student Version)

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MODULE : MATHEMATICS 2

(Diploma)

GOALS:

The goal of this course is to enhance students’ understanding and knowledge in Mathematics for technical applications.

LEARNING OUTCOMES :

After completing this course successfully, students should be able to :

1. Apply Pyhtagoras Theorem to solve problems in geometry for a right

angled triangle.

2. Compute the distance, mid-point and slope between two coordinates

points in Cartesian Coordinate System.

3. Derive the equation of lines based on the linear equation, y = mx + c.

4. Solve the linear equations problems with one unknown.

Solve the system of linear equations by using the inverse matrix method for 2 by 2

matrices.

LEARNING CONTENT:

This module consists of :

Unit Title Hours

Unit 1 INDICES, LOGARITHM AND SURDS 6

Unit 2 MATRICES 10

Unit 3 TRIGONOMETRY 10

Unit 4 FUNDAMENTAL STATISTICS 9

Unit 5 STATISTICS 10

ASSESSMENT :

Assignments : 10 %

PBL/Presentations : 30 %

Quizzes : 10 %

Tests : 20 %

Final Examinations : 30%

REFERENCES :

1. Abd. Wahid Md Raji & Rakan-rakan. (2000). Matematik Asas - Jilid I, Jabatan Matematik, Fakulti Sains, UTM.

2. Abd Wahid Md Raji & Rakan-rakan.(2000) Matematik Asas- Jilid II, Jabatan Matematik, Fakulti Sains, UTM.

3. Barnett, R. A., Ziegler, M.R. and Byleen, K.E. (2000). College Algebra with Trigonometry, 7th Ed. New York: Mc Graw Hill.

4. Kaufmann, J.E. and Schwitters,K.(2000). Algebra for College Students, 6th Ed. Pacific Grove: Brooks/Cole Thomson Learning.

5. Quek Suan Goen, Leng Ka Man & Yong Ping Kiang (2004). Mathematics STPM. Federal Publications, Selangor.

6. Woodbury, G.(2004). An introduction to Statistics. Thomson Learning.

UNIT 1

INDICES, LOGARITHM AND SURDS

1.1 Introduction

In this topic students will be exposed to the rules of fundamental algebra in

solving problems dealing with indices, logarithms and surds. At the end of it,

students will see the connection between them.

Objectives

At the end of the topic, students should be able to:

use the laws of indices in solving problems

use the laws of logarithms in solving problems with general based logarithms,

common logarithms and natural logarithms.

rationalize the numerator or the denominator of surds and solve problems

concerning surds.

1.2 Indices and Laws of Indices

1.2.1 Indices

(A) Positive integer indices

If a is a non-zero numeral and n is a positive integer, then means a is

multiplied by itself n times and is called a to the power of n.

n times

In , a is the base and n is the index.

Example 1.1

Find the value of

(a) . (b) .

(c) . (d) .

(B) Zero index

Any number with zero indexes equals to one. For instance,

Example 1.2

Find the value of each of the following without using a calculator.

(a) . (b) ( . (c) .

(C) Negative indices

is the reciprocal of

Example 1.3

Find the value of each of the following and verify your answers using a

calculator.

(a) . (b) . (c) .

(D) Fractional Indices

1. is the nth root of a, that is

2. is the nth root of , that is

Example 1.4

Find the value of

(a) . (b) .

(c) . (d) .

Example 1.5

Find the value of

(a) . (b) .

1.2.2 Laws of Indices

Example 1.6

Find the value of

(a) . (b) .

(c) . (d) .

Example 1.7

Simplify and evaluate each of the following.

(a) . (b) .

Example 1.8

Simplify each of the following.

(a) . (b) .

(c) . (d) .

Example 1.9

Simplify each of the following.

(a) . (b) .

(c) . (d) .

Practice 1.1

1. Find the value of

(a) . (b) .

Solution

(a) means _____ is multiplied by itself ________ times.

(b) means _____ is multiplied by itself ________ times.

Hence, .

(a) . (b) .

Solution

(a) . (b) . (c) .

Solution

(a) . (b) . (c) . (d) .

Solution

(a) Use the laws of indices. What should be done to the index if indices with the

same base are multiplied?

(b) By applying the laws of indices, what should be done to the index?

(c) Do you know which laws to apply? Check the base and the operation involve!

(d) Refer to the laws of indices. What should be done to the index?

5. Simplify each of the following.

(a) . (b) . (c) . (d) .

Solution

(a) Same base multiplied, so the indices should be ……..…

(b) Same base divided, so the indices should be ………….

(c) What should be done to the index?

(d) Please refer to the laws of indices.

1.3 Logarithms and Laws of Logarithms

1.3.1 Definition of Logarithm

If a is a positive number and then

is read as ‘logarithm of N to the base of a’.

Note : 1. .

Example 1.10

Express each of the following in logarithmic form.

Example 1.11

Express each of the following in index form.

1.3.2 Logarithm of a number

1. Logarithms to the base of 10 are called common logarithms. The values

of common logarithms can be determined by using a scientific

calculator.

2. Logarithms to the base of e are called the natural logarithms. The natural

logarithms of x is usually written as ln x instead of .

Example 1.12

By using a scientific calculator, find the value of

(a) . (b) .

Note : 1. Logarithm of negative numbers is undefined.

2. Logarithm of zero is undefined.

Example 1.13

Find the value of

(a) . (b) .

Example 1.14

Find the value of x in each case.

(a) . (b) .

(c) . (d) .

1.3.3 Laws of Logarithm

There are three basic laws of logarithms.

Example 1.15

Given that and , find the value of each of the

following without using a calculator.

(a) . (b) . (c) .

Example 1.16

Evaluate the following without using a calculator.

(a) . (b) .(c) .

Example 1.17

Given that and find the value of .

Example 1.18

Given that and , express in terms of x and y.

Practice 1.2

1. Express each of the following in logarithmic form.

(a) . (b) . (c) .

Solution

(a) If , then .

(b) If , then

(c) If , then .

2. Express each of the following in index form.

(a) . (b) .

Solution

(a) If , so .

(b) If , so .

3. Evaluate the following without using a calculator.

(a) . (b) . (c) .

Solution

(a) Write 81 in the index form. Can you determine the

(b) Write 2 as the fractional indices of 8. What should be the index?

(c) . Write 100 in index form with base ______.

4. Find the value of without using a calculator.

Solution

1.3.4 Change of Base of Logarithms

Let So .

By taking logarithm to base c on both sides, we have

As , hence

Note : We can find the logarithm of a number by changing the base of the

logarithm to any appropriate base.

Example 1.19

Find the value of each of the following without using a scientific calculator.

(a) . (b) . (c) .

Example 1.20

Find the value of

1.3.5 Solving problems involving the change of base and laws of

logarithms

Example 1.21

If and , express each of the following in terms of m and n.

(a) . (b) .

Example 1.22

Simplify .

1.3.6 Equation Involving Indices and Logarithms

(A) Solving equations involving indices

Steps in solving :

- simplify the algebraic expressions on both sides of the equation

- express the expressions in terms of the same base or the same index

- equate the base or the indices accordingly

For example,

if , then m = n.

if , then a = b.

Example 1.23

Solve the following equations.

(a) . (b) . (c) .

(B) Solving equations involving logarithms

Example 1.24

Solve the following equations.

Practice 1.3

1. Solve for x if .

Solution

Apply log with base 10 to both sides of the equation.

What happen if you use log with base e, the natural log?

Does it give the same answer? Please try….

2. Solve the following equations.

Solution

So, the answers for x are : ……………………..

Solution

Arrange the terms with x,

Calculator!

Eliminate the denominator,

Thus, the answers for x are : ……………………..

3. Determine the value of x if

Solution

You have to solve it by substituting to the equation.

Please continue on the steps of solving. You are correct if the answers are :

x = 2 and x = 2.322.

1.4 Surds and Laws of Surds

The expression is a radical, the number a is the radicand, and n is the index of

the radical. The symbol is called a radical sign.

1.4.1 Laws of Surds

The three listed laws in the next chart are true for positive integer’s m and n,

provided the indicated roots exist.

Example 1.25

Simplify the following expressions by using the laws of surds.

(a) . (b) . (c) .

1.4.2 Rationalizing the Numerator and Denominator of Surds

If the numerator and denominator of a quotient contains of the form , with k <

n and a > 0, then multiplying numerator and denominator by will eliminate

the radical.

Example 1.26

Rationalize the denominator or the numerator.

Practice 1.4

1. Rationalize the numerator.

(a) . (b) .

Solution

(a) The numerator and the denominator should be multiplied by ………..

= ………..

(b) The numerator and the denominator should be multiplied by ……………….

Hence,

= …………………….

2. Rationalize the denominator.

(a) . (b) . (c)

Solution

(a) The …………………….. and the ……………………… should be multiplied

by ……………..

So, the answer is .

(b) The …………………….. and the ……………………… should be multiplied

by ……………..

Hence,

(c) Try this on your own!

The …………………….. and the ……………………… should be multiplied

by ……………..

You got it right if the answer is .

EXERCISE

1. Find the value of the following.

(a) . (b) .

(c) . (d) .

(e) . (f) .

(g) . (h) .

(i) . (j) .

(a) . (b) .

(c) . (d) .

(e) . (f) .

(g) . (h) .

(i) . (j) .

3. Simplify and evaluate each of the following.

(a) . (b) .

(c) . (d) .

(e) . (f) .

4. Simplify.

(a) . (b) .

(c) . (d) .

(e) . (f) .

5. Express each of the following in logarithmic form.

(a) . (b) .

(c) . (d) .

6. Convert each of the following to index form.

(a) . (b) .

(c) . (d) .

7. Find the value of x in each case.

(a) . (b) .

(c) . (d) .

(e) . (f) .

8. Given that and find the value of the following

without using a calculator.

(a) . (b) .

(c) . (d) .

(e) . (f) .

9. Simplify the following logarithmic expressions to the simplest form.

(a) . (b) .

(c) . (d) .

(e) . (f) .

11. Given that and , find the value of each of the

following without using a scientific calculator.

(a) . (b) .

12. If , express each of the following in term of t.

(a) . (b) .

(c) . (d) .

(e) . (f) .

13. If and , express each of the following in terms of p and q.

(a) . (b) .

(c) . (d) .

15. Rationalize the denominator.

(a) . (b) .

(c) . (d) .

(e) . (f) .

16. Rationalize the numerator.

(a) . (b) .

Activity

1. Savings account

One of the oldest bank in Malaysia is the Standard Chartered Bank. If RM 200

had been deposited at that time into an account that paid 4% annual interest,

then 180 years later the amount would have grown to dollars.

Approximate this amount to the nearest cent.

2. Length of a halibut

The length-weight relationship for Pacific halibut can be approximated by

, where W is in kilograms and L is in meters. The largest

documented halibut weighed 230 kg. Estimate its length.

3. Weight lifter’s handicaps

O’Carroll’s formula is used to handicap weight lifters. If a lifter who weighs b

kg lifts w kg of weight, then the handicapped weight W is given by

Suppose two lifters weighing 75 kg and 120 kg lift weights of 180 kg and 120

kg lift weights of 180 kg and 250 kg, respectively. Use O’Carroll’s formula to

determine the superior weight lifter.

Answer

1. RM 232 825.78

2. 2.82m

3. The 120 kg lifter

Unit1 Exp,Log,Surd MATH2(D) Ikbn (Student Version)

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