First Edition EDIT O RS MAISURAH SHAMSUDDINSITI BALQIS MAHLANNORAZAH UMARUNIT PENULISAN DAN PENERBITANJSKMDepartment of Computer and Mathematical Sciences Universiti Teknologi MARA Cawangan Pulau Pinang, Malaysia
INTERMEDIATE MATHEMATICS FOR STEM C Editors Maisurah Shamsuddin, Universiti Teknologi MARA Cawangan Pulau Pinang, Malaysia Siti Balqis Mahlan, Universiti Teknologi MARA Cawangan Pulau Pinang, Malaysia Norazah Umar, Universiti Teknologi MARA Cawangan Pulau Pinang, Malaysia Copyright@2023 by Penerbitan JSKM Department of Computer and Mathematical Sciences, Universiti Teknologi MARA Cawangan Pulau Pinang, 13500 Permatang Pauh, Pulau Pinang, MALAYSIA All rights reserved. No parts of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying or otherwise, without the prior written permission of the publishers. Perpustakaan Negara Malaysia Data-Pengkalan-dalam-Penerbitan ISBN : 978-967-25608-9-0 Printed in Malaysia
PREFACEIntermediate Mathematics for STEM C is written specifically for the SciencePre-Diploma students (STEM C and Literature) at Universiti Teknologi Mara. It contains all the topics stipulated in the syllabus. This book covers five basic subjects at KSSM level. The topics are Functions, indices and logarithms, inequalities, circular measures and solving triangles. For each subtopic, examples and detailed solution processes are presentedtoaid student understanding. Each subtopic ends with a simple exercise calledLet's Master Mathematics which serves as a reinforcement exercise andteststhe student's understanding of the topic discussed. Each chapter ends withadditional exercises that offer a variety of exam-style questions based ontheconcepts and theories covered. We would like to thank everyone who has contributed directly or indirectlytomaking this book a reality. We hope this book will be of great help to studentsin mastering mathematical knowledge as well as preparing themfor theirexaminations.
ii Preface iContents iiChapter 1: Function 1.1 Introduction 11.2 Basic Operation of Functions 61.3 Composite Function 101.4 Inverse Function 15Exercise & Answers 19Chapter 2: Index and Logarithm2.1 Index Function 212.2 Logarithms Function 302.3 Equations Involving Index and Logarithms 43Exercise & Answers 55Chapter 3: Inequality 3.1 Introduction to Inequality notations, Range and Number Lines 603.2 Laws of Inequality: Addition, Subtraction, MultiplicationandDivisions 633.3 Solving Inequality : Linear Inequality 663.4 Solving Inequality: Quadratic and Rational Inequality 693.5 Solving Inequality : Absolute Value (Linear) 73Exercise & Answers 75Chapter 4: Circular Measure 4.1 Introduction of Converting Measurement of Angle 784.2 Arc Length and Chord of a Sector 814.3 Area and Perimeter of Sector and Segment 85Exercise & Answers 89
iii Chapter 5: Solutions of Triangle 5.1 Right - Angled Triangle Solutions. 965.2 Sine Rule and Cosine Rule. 1005.3 Area and Perimeter of Triangle. 105Exercise & Answers 110
INTERMEDIATE MATHEMATICS (STEM C) 1CHAPTER 1 FUNCTIONS Objectives At the end of this chapter the student should be able to Understand the concept of a function. Understand the domain and range of functions. Understand the terms one-to-one function and many-to-one function. Solve function for basic operation, composite function and inverse function. 1.1 INTRODUCTION OF FUNCTION A relation maps elements from one set to the elements of another set. To help understand this definition, look at this mapping from set A into set B. Therelation consists of the ordered pairs ( a, d ) and ( b, c ). Domain : { a, b } Codomain : { c, d, e } Range : { c, d } A relation can be represented using a) arrow diagram b) graph c) ordered pairs A relation can be classified as One-to-one Many-to-one A B a b c d e Domain Codomain Domain Codomain
INTERMEDIATE MATHEMATICS (STEM C) 2Codomain physical state paper ice water solid liquid gas number greater than 2 8 3 4 10 square of 2 5 7 4 25 49 type of number even odd 2 3 4 5 A function is a special relation where every object in the domain has one and only oneimage in the codomain. Therefore, only one-to-one relations and many-to-onerelations are functions. Example 1 Determine whether each of the following relation is a function. Give reasons for your answer. a) b) c) d) Solution: a) Function (many-to-one). b) Not a function (one-to-many). c) Function (one-to-one). d) Function (many-to-one). Codomain Codomain Codomain Domain Domain Domain Domain
INTERMEDIATE MATHEMATICS (STEM C) 3Example 5 Determine if each of the following are functions. a) 2 y x 2 Solution: Function! Because whatever value of x we put into the equation, there is only one possible value of y when we evaluate the equation at that value of x. b) 2 y x 2 Solution: Not a function! For example, when we put x 2 into the equation, there are two possible values of y we could use. 2 2 2 y 2 2 4 y 4 y 4 y 2 We could use y 2 and y 2 .Since there are two possible values of y that we get from a single x, this equation isn’t a function. NOTE! All functions are relations but all relations are not functions. Relations Functions
INTERMEDIATE MATHEMATICS (STEM C) 4Determine whether each of the following relation is a function. Give reasons for your answer. 1. y x 2 2. 3. 4. Answer: Reason: Answer: Reason: Answer: Reason: Answer: Reason: 2 3 4 4 5 6 Anna Siti Ayu Sara Maria Murni Daughter Mother Ali Hasif Sazzy Alia Misya A B AB O People Blood Type Siva Ismail Ilman Adra Muaz Putri Teacher Students
INTERMEDIATE MATHEMATICS (STEM C) 5A relation between set A and set B is represented by the ordered pairs given below: 1,5 2,6 3,6 4,7 a) Describe the relationship above using the arrow diagram. b) State the type of relationship. Solution: A relation between set A and set Bisrepresented by the ordered pairs givenbelow: a b c d c f g h, , , , a) State whether the relationship aboveisa function or not. Give a reason. b) Find the image of c. Solution: A relation between set A and set B is represented by the ordered pairs given below: 2,4 3,9 4,16 5,25 a) Describe the relationship above using the arrow diagram. b) Is it a function?Why? Solution: A relation between set A and set Bisrepresented by the ordered pairs givenbelow: 1,3 2,6 1,5 4,6 a) Describe the relationship above usingthe arrow diagram. b) Find the image of 2 and the object of 6. Solution:
INTERMEDIATE MATHEMATICS (STEM C) 61.2 BASIC OPERATION OF FUNCTIONS Given any two functions f and g, can be added, subtracted, multiplied and dividedinanatural way to form new function f + g, f – g, fg and f / g. Generally the following rules applies: Example 1 Given f(x) 5x 1 and g(x) 4x , find f gx.Then evaluate the sumwhen x3. Solution: 93 1 28 9x 1 f g x f x g x 5x 1 4x Example 2 Given f(x) 2x 7 and g(x) 3x , find f gx.Then evaluate the sumwhen x2. Solution: f g x f x g x 2x 7 3x 7 x 7 2 5
INTERMEDIATE MATHEMATICS (STEM C) 7Example 3 Given f(x) 6x 5 and g(x) 2x , find fgx.Then evaluate the sum when x4. Solution: 2 2 fg x f x g x 6x 5 2x 12x 10x 12 4 10 4 232 Example 4 Given f(x) 3x 1 and g(x) x , find f / gx. Solution: x 1 3 x 3x 1 g x f x x g f Example 5 Given f(x) 2x 3 and g(x) x 4 , find 5(f g)(10) . Solution: 5 f g x 5 f(x) g(x) 5 2x 3 x 4 5 2x 3 x 4 5 3x 7 15x 35 5 f g (10) 15(10) 35 150 35 185
INTERMEDIATE MATHEMATICS (STEM C) 8Example 6 Given f(x) x 8 and g(x) 3x 4 , find (f g)(5) . Solution: f g x f(x) g(x) x 8 3x 4 x 8 3x 4 2x 4 f g (5) 2(5) 4 10 4 6 Example 7 Given that f x x 3 1 and g x x 2 5 . Find a) g 3 b) the value of a if f a g 2 2 (3) Solution: a) g(3) 2 (5)(3) 13 b) f a g 2 2 (3) 3(a 2) 1 2( 13) 3a 6 1 26 3a 21 21 a 3 7
INTERMEDIATE MATHEMATICS (STEM C) 91. Given that f(x) 3x 2 and g(x) 1 5x . Find a) f(2) b) the value of a if 3 2 ( 5) f g a Solution: 2. Given that f(x) 4x 3 and g(x) 2x 5 . Find a) g(1) b) the value of b if f b g 2 (1) 3Solution: 3. Given that f(x) 6 x and x g(x) . 2 Find a) g(8) b) the value of a if f a g 2 1 3 (8) Solution: 4. Given that f(x) x 7 and g(x) 2x 7 . Find a) f(4) b) the value of a if 2 4 3 ( 2) f g a Solution: Let's Master Mathematics!
INTERMEDIATE MATHEMATICS (STEM C) 101.3 COMPOSITION OF FUNCTIONS A composition function is a composition of two functions. Given two function f andgdenoted as f gis fgx defined as f g fgx In general, fgx gfx, fg and gf are different functions. The composition of a function f and itself can be written as: Example 1 Given that f(x) 2x 1 and ,x 1. x 1 2x g x Find a) fg(x) b) gf(x) A B C x y z f(x) g(y) gf(x)
INTERMEDIATE MATHEMATICS (STEM C) 11Solution: Example 2 Given that f(x) 2x 5, find a function g such that fgx 6x 1. Solution: 3x 2 2 6x 4 g x 2g x 6x 1 5 2g x 5 6x 1 fg x 6x 1 x 1 3x 1 x 1 4x x 1 x 1 4x x 1 1 x 1 4x 1 x -1 2x 2 x 1 2x a) fg x f x 1 2x 1 2x 2 4x 2 2x 1 1 4x 2 2x 1 1 2 2x -1 b) gf x g 2x 1
INTERMEDIATE MATHEMATICS (STEM C) 12Example 3 Given that , x 2 x 2 x f(x) and ,x 1 1 x 1 2x g x . Find a) fg(4) b) gf(4) Solution: 3 3 2 3 f 3 3 9 f 1 4 1 2 4 f 1 x 1 2x a) fg 4 f 7 3 2 3 4 3 2 1 3 4 1 3 2 1 3 2 1 2 3 2 g 4 2 4 g x 2 x b) gf 4 g
INTERMEDIATE MATHEMATICS (STEM C) 13The functions f and g are defined as f x x 2 1 and g x x 2 3 . Find a) fg x b) fg 1 c) The value of x if fg x 27 Solution: Given the function f(x) x 2 andg(x) mx 4 , find the value of msuchthat gf( 2) 8 . Solution:
INTERMEDIATE MATHEMATICS (STEM C) 14The functions f and g are defined as f x x 2 and 2 g x x . Find a) gf x b) gf 3 c) The value of x if gf x 16 Solution: Given the function x 4f(x) 10and2 g(x) x s , find the value of s suchthat fg( 1) 5 . Solution:
INTERMEDIATE MATHEMATICS (STEM C) 151.4 INVERSE FUNCTIONS An inverse function is a function which inverse the mapping between two sets, AandB. The function which maps set B onto set A is called the inverse function of f andit isdenoted by 1 f . If x is mapped onto y, then fx y and f y x 1 . So, only one-to-onefunction has inverse function. Example 1 Given that ,x 3 x 3 2x 3 f(x) . Find f 3 1 . Solution: 6 3 2 3 3 3 f 3 when x 3 x 2 3 3x f x y 2 3 3y x x y 2 3 3y yx 2x 3 3y yx 3y 2x 3 x 3 2x 3 Let y 1 1 A B x y f f -1
INTERMEDIATE MATHEMATICS (STEM C) 16Example 2 Given that ,x 4 x 4 1 x f(x) . Find f 3 1 . Solution: 2 11 3 1 1 4 3 f 3 when x 3 x 1 1 4x f x y 1 1 4y x x y 1 1 4y yx x 1 4y yx 4y 1 x x 4 1- x Let y 1 1 Example 3 Given that ,x 3 x 3 x p f(x) and f 3 4 1 . Find the value of p. Solution: 1 x p Let y x 3 yx 3y x p yx x p 3y x y 1 p 3y p 3y x y 1 p 3x f x x 1 -1 1 when f 3 4 p 3 3 f 3 4 3 1 p 9 8 p 17
INTERMEDIATE MATHEMATICS (STEM C) 17Example 4 Given that ,x 1 x 1 3 f(x) and gx x 4. Find a) 1 gf b) 1 1 f g Solution: x 4 x 1 gf x y 4 y 1 x x y 4 y 1 yx 4x 1 y yx y 4x 1 x 1 4x -1 Let y Therefore x 1 4x 1 x 1 3 4 x 1 4 x 1 3 x 1 3 a) g 1 x 3 3x f x y 3y x yx 3 y yx y 3x 13 Let y b) f x y 1 -1 -1 1 1 1 1 Let g x y y x 4 x y 4 g x x 4Therefore, f g f x 43 x 4 x 4x 1x 4
INTERMEDIATE MATHEMATICS (STEM C) 18Given the function g(x) 3x 2 . Find the inverse function 1 g 5 . Solution: Given that f x x 2 5 3. Find 1 f x . Solution: Given the function g(x) 5x 4 . Find the inverse function 1 g 2 . Solution: Given that f(x) 2x 1, and gxx4. Find 1 gf 9 . Solution:
INTERMEDIATE MATHEMATICS (STEM C) 19EXERCISE 1. Given that 2 x 3 f x and 1 2x 3x 1 g x . Determine a) g fx b) g x 1 2. Given that x 1 1 f x and gx x 2 2 . Determine a) f gx b) f x 1 3. Given that fx 3x 2 and 3x 3x 2 gf x , find gx. 4. Given that fx 6x 12 and gx x 5 . Find the value of p if fp 4gp. 5. Given that fx 2 3x and 3 2x 1 g x . Find a) the value of x if fgx x b) the value of x if gfx 2 6. Given that fx 2x 1 and fgx 2x 3 2 . Determine a) gx b) the value of x ifg f0 g x 1 7. Given that ,x 0 x 2 f x 4 and 3 1 ,x 3x 1 12x f h x . Find a) hx b) h x 2 c) the value of x such that hx 1 x 8. Given that fx 3x 5 and f gx x 2x 5 1 2 . Find a) f x 1 b) gx c) the value of x if f x x 2
INTERMEDIATE MATHEMATICS (STEM C) 209. Given that gx 3x 5 and ,x 5k x 5k kx 4 f x . If f2 6 , find a) the value of k b) f x 1 c) f gx 10. Given that fx x 11 and x 5 x 1 fg x . Find a) gx b) fg x 1 Answers 1. a) 8 2x 3x 7 b) 3 2x x 1 2. a) x 3 1 2 b) x 1 x 3. x 2 x 4. 5 16 p 5. a) x 1 b) 2 3 x 6. a) x 2 2 b) x 11 7. a) 2 3x 1 b) 4 9x 5 c) x 2 8. a) 3 x 5 b) 3x 6x 20 2 c) 2 5 x 9. a) 4 1 k b) 4x 1 16 5x c) 12x 25 3x 11 10. a) x 5 12x 56 b) x 1 1 5x
INTERMEDIATE MATHEMATICS (STEM C) 21CHAPTER 2 INDEX AND LOGARITHMSObjectives At the end of this chapter the student should be able to: simplify the exponentials and logarithmic expressions. write an index expression in logarithmic form or vice versa. solve equations involving indices and logarithms using certain rules. 2.1 INDEX FUNCTION What are indices? An index is a small number that tells us how many times a termhas beenmultiplied by itself. The plural of index is indices. Any expression written as a n is defined as the variable a raised to the power of the number n. The following notation shows the product of n factors of a: n a a a a a.... a Example 1 4 5 5 is the base and 4 is the index We can read this as ‘5 to the power of 4’. Another way of expressing 4 5 is 5 5 5 5 625 n factors
INTERMEDIATE MATHEMATICS (STEM C) 22Example 2 Evaluate each of the following: a) 5 3 b) 3 10 c) 4 1 2 d) 3 ( 2) Solution: a) 5 3 3 3 3 3 3 243 b) 3 10 10 10 10 1000 c) 2 1 1 1 1 2 2 2 4 d) 3 ( 2) ( 2) ( 2) ( 2) 8 2.1.1 Types of Indices Zero Index If a is nonzero real number a 0 , then 0 a 1 Example 3 a) 0 5 b) 0 ( 240) c) 0 (7.48) Solution: a) 0 5 1 b) 0 ( 240) 1 c) 0 (7.48) 1
INTERMEDIATE MATHEMATICS (STEM C) 23Negative Index If a is any real number where a 0 and n is an integer, n n 1 a a n n n n n a 1 b bb aaab n n n 1 1 a a 1 a n n n n n n 1 aa b ab b Example 4 State the following using positive exponent. a) 2 4 b) 3 4 3 2 c) 5 3 s t d) 2 96 mSolution: a) 2 4 2 1 1 4 16 b) 3 4 3 2 3 4 1 1 1 3 2 27 16 432 c) 5 3 st 5 3s t d) 2 9 9 36 6 m m
INTERMEDIATE MATHEMATICS (STEM C) 24Rational Index If a is any real number where a 0 , m and n is positive integer, then 1 n n a a m mn n m n a a a 1 n a is called the principal nth root of a. The symbol is called a radical, n is called theindex, and a is called the radicand. Example 5 Evaluate the following: a) 1 3 8 c) 4 125 5 b) 1 1 2 256 d) 3 2 8 Solution: a) 1 3 3 8 8 2 c) 4 4 5 4 125 125 3 815 b) 1 1 1 2 256 256 1 16 d) 2 2 2 3 3 3 1 127 27 27 2 1 13 9
INTERMEDIATE MATHEMATICS (STEM C) 25Evaluate each expression: 1. 4 2 2. 4 2 3. 1 8 3 27 4. 2 1 4 5. 3 0 81 4 6. 2 125 3
INTERMEDIATE MATHEMATICS (STEM C) 262.1.2 Law of Index If a and b are nonzero real numbers and m and n are integer, then m n m n a a a Product rule of power n m mn a a Power of a product rule m m n n a a a Quotient rule of power m m m ab a b Power of product rule m m m a a b b Power of Quotient rule Example 1 Simplify a) 2 3 7 4 4 4 4 b) 5 2 3 2a b c) 2 4 2 4x 2y d) 3 4 2 3 8x y 12x y e) 3 2 2 2 5m n m n Solution: a) 2 3 7 2 3 7 1 1 1 4 4 4 4 4 4 4
INTERMEDIATE MATHEMATICS (STEM C) 27b) 10 5 2 3 10 15 15 32a 2a b 32a b b c) 2 4 2 4 2 8 8 2 2 2 2 4 4 4x 4 x 16x 4x 2y 2 y 4y y d) 3 4 3 2 4 3 5 2 3 8x y 2 2 x y x y 12x y 3 3 e) 3 2 3 2 3 3 2 2 2 2 2 5m n 5 m n m n m n 3 2 3 3 6 4 3 2 2 2 2 2 5 m n 125m n 125m nm n Example 2 Simplify a) 2 3 5 7 4 3 4p q 9p q 6p q b) 4 3 5 2 3 5 p p q q Solution: a) 2 3 5 7 7 10 4 3 4 3 4p q 9p q 36p q 6p q 6p q 7 4 10 3 3 7 6p q 6p q b) 4 3 5 2 20 6 3 5 12 15 p p p p q q q q 20 15 12 6 20 15 12 6 20 6 15 12 14 3 p q q p p q q p p q p q
INTERMEDIATE MATHEMATICS (STEM C) 28n 1 a n m m m ab a b n 1an 1 n n a a m m n n a a a Example 3 Simplify the following using the rules of indices. State the answers in positive indices. a) 1 2 3 2 3 2 3 2 a b a b b) 1 2 2 3 6 3 64x y c) 22 2a b Solution: a) 1 2 3 2 3 3 1 2 3 2 2 3 2 3 2 a b a b a b a b 3 2 1 2 2 3 3 a b 5 5 6 3 a b 5 6 5 3 a b b) 1 1 1 2 2 2 2 3 1 2 2 3 6 3 3 6 64x y 64 x y 1 1 1 2 3 2 2 2 6 3 64 x y 1 1 3 3 3 8x y 1 1 1 1 3 3 3 3 3 8 x y 1 9 2xy 1 9 2x y c) 2 2 2 2 2 2 1 1 a b a b 2 2 2 2 2 4 4 2 2 2 b a b a b a b a
INTERMEDIATE MATHEMATICS (STEM C) 29Simplify the following using the rules of indices. State the answers in positive indices. 1. 3 2 1 3 m n m n 2. 2 3 4 3 ab a b a b 3. 4 3 5 4 3 2 m m n n 4. 1 n 5n 6 9n 3 27 3 5. 1 5 1 2 3 5 25a b 16a b 6. p q q p r 2 2 3 2 4 6 2 4
INTERMEDIATE MATHEMATICS (STEM C) 302.2 LOGARITHM FUNCTION 2.2.1 Logarithmic Function and Notation The function a log x is defined as the inverse of the exponential function x a. In other words, where a, x, y > 0, a ≠ 0 Example 1 a) If 3 log 81 x then 3 x = 81 b) If 4 log 64 x then x 4 64 c) If 2 1 log x 256 then x 1 2 256 d) If x 64 1 log8 then 6418 x 2.2.2 Basic Properties of Logarithmic Function If a is a positive number and a 1 , then: Properties Example 1. a log 1 0 1. 3 log 1 0 2. 100 log 1 02. a log a 1 5 log 5 1 3. x a log a x 1. x 4 log 4 x 2. 2 3 log 3 2 4. x a a log b xlog b 1. 3 3 3 log 4 3log 4 a log x y is equivalent to y x a Common logarithm (base 10) 10 log x log x Natural logarithm (base e) e log x ln x
INTERMEDIATE MATHEMATICS (STEM C) 31Let's Master Mathematics! Without using a calculator, simplify the following. 1) 8 log 1 2) 4 log 4 3) 3 5 log 5 4) 5 2 log 4 1.2.3 Rules of Logarithmic Function 1. Power Rule If x is a positive number and n is any rational number, then n a a log x nlog xExample 1 Evaluate without using a calculator: a) 3 2 log 2 b) 3 log 729 c) 5 log 125 Solution a) 3 2 2 log 2 3log 2 3 1 3
INTERMEDIATE MATHEMATICS (STEM C) 32b) 6 3 3 log 729 log 3 6log 33 6 1 6 c) 1 2 5 5 log 125 log 125 1 3 2 5 log 5 3 2 5 log 5 5 3 log 5 2 3 1 2 3 2 Example 2 Without using a calculator, simplify 5 3 2 log 25 log 81 log 32 . Solution 5 3 2 2 4 5 5 3 2 5 3 2 log 25 log 81 log 32 log 5 log 3 log 2 2log 5 4log 3 5log 2 2(1) 4(1) 5(1) 1
INTERMEDIATE MATHEMATICS (STEM C) 33Let's Master Mathematics! Without using a calculator, evaluate the following: a) 5 4 log 4 b) 5 log 625 c) 3 log 243 d) 2 log 8 e) 3 5 4 log 27 log 1 log 64 f) 4log 8 log 9 log 52 3 5
INTERMEDIATE MATHEMATICS (STEM C) 342. Product Rule If x and y are positive numbers, then a a a log xy log x log y . In other words, thelogarithm of a product is the sum of the logarithms. Example 1 Write each expression in terms of the logarithms of x, y and z. a) b log xyz b) 3 2 b log x y z Solution a) b b b b log xyz log x log y log z b) 3 2 3 2 b b b b log x y z log x log y log z Example 2 Evaluate without using a calculator: a) 6 6 log 3 log 2 b) 2 2 log 50 log 4 c) 3 2 log 9 log 32Solution a) log 3 log 2 log 3 2 6 6 6 6 log 6 1 CAUTION a a a log x y log x log y or a a log x y log xy Apply properties: a log a 1
INTERMEDIATE MATHEMATICS (STEM C) 35c) 2 5 3 2 3 2 log 9 log 32 log 3 log 2 2 log 3 5 log 2 3 2 2 1 5 1 7 Let's Master Mathematics! Evaluate without using a calculator: a) 5 5 log 5 log 25 b) 2log 16 log 644 4 b) log 64 log 8 log 64 8 2 2 2 2 log 512 9 2 log 2 9 log 2 2 9 1 9 Apply Product Rule of Logarithm Apply basic properties of logarithm
INTERMEDIATE MATHEMATICS (STEM C) 36c) 3 2 log 27 2log 64 d) 3log 16 log 814 3 3. Quotient Rule If x and y are positive numbers, then a a a x log log x log y y . In other words, thelogarithm of a quotient is the different of the logarithms. Example 1 Write each expression in terms of the logarithms of x, y and z. a) b x log z b) b 2 xy log z CAUTION a a a log x y log x log y or a a x log (x y) log y
INTERMEDIATE MATHEMATICS (STEM C) 37Solution a) b x log z b b log x log z b) 2 b 2 b b 2 b b b xy log log (xy) log z z log x log y log z Example 2 Evaluate without using a calculator: a) 4 4 log 64 log 16 b) 3 7 log 81 log 343 c) 2 2 log 8 log 64Solution a) 4 4 4 64 log 64 log 16 log 16 4 log 4 1 c) 2 2 log 8 log 64 2 2 1 2 8 log 64 1 log 8 log 8 Apply Product and Quotient Rule for Logarithms Apply properties: a log a 1 b) 4 3 3 7 3 7 log 81 log 343 log 3 log 7 4log 3 3log 7 3 7 4(1) 3(1) 1 Apply Quotient Rule of Logarithm
INTERMEDIATE MATHEMATICS (STEM C) 38 3 1 log 2 2 1 3 log 2 2 3 log 2 2 a 3 1 3 Example 3 Simplify the following expression as a single logarithm. 2 10 10 log 3x log 4y 1 Solution 2 2 10 10 10 10 10 log 3x log 4y 1 log 3x log 4y log 10 10 10 10 10 3x log log 10 4y 3x log 10 4y 15x log 2y Apply properties: a log a 1
INTERMEDIATE MATHEMATICS (STEM C) 39Let's Master Mathematics! Evaluate without using a calculator: a) 2log 5 2log 4 log 50 2 2 2 b) 1 log 21 log 7 3 3 c) 3 4 5 1 1 2log log log 625 9 4 d) 2log 8 4log 7292 3
INTERMEDIATE MATHEMATICS (STEM C) 40e) Simplify 2 5 5 5 log x log y log 3y as a single logarithm. f) Simplify 2 3log ab log a b 14 3 as a single logarithm. 1.2.4 Change of Base If a, b and x are real numbers, then: Example 1 Find the value for each of the following. a) 3 log 6 b) 5 log 7 Hint: 1) Change the logarithmic function into base 10. 2) Calculate using calculator.
INTERMEDIATE MATHEMATICS (STEM C) 41Solution a) 10 3 10 log 6 log 6 log 3 0.7782 0.4771 1.6309 Example 2 Given a log 3 0.641 and a log 5 1.456 . Find the value of 2 a a log 15 . Solution 2 2 a a a a a a a a log log a log 15 15 2log a log 5 3 2 1 log 5 log 3 2 1.456 0.641 2 2.097 0.097 b) 10 5 10 log 7 log 7 log 5 0.8451 0.6990 1.2091
INTERMEDIATE MATHEMATICS (STEM C) 42Let's Master Mathematics! a) Evaluate 3 log 12 to four decimal places. b) Evaluate 0.2 log 4.5 to four decimal places. c) Find the value of 8 log 14 if given 2 log 7 2.81 . d) Given that 3 log 2 0.6309 and 3 log 5 1.465 . Find the value of 25 log 10 .
INTERMEDIATE MATHEMATICS (STEM C) 431.3 EQUATIONS THAT INVOLVING INDICES AND LOGARITHMS 2.3.1 Solving equation involving indices. Equations involving exponents can sometimes be solved using the following properties of exponents. Expressions with the same base but different exponents can only be the sameif and only if the exponents are the same. Example 1 Find the value of t in the equation 2t 6 3t 4 8 8 . Solution: 2t 6 3t 4 8 8 2t 6 3t 4 2t 3t 4 6 t 10 t 10 Example 2 Find the value of s in the equation 2s 1 6 27 3 . Solution: 2s 1 6 27 3 2s 1 3 6 6s 3 6 3 3 3 3 Therefore, 6s 3 6 6s 9 3 s 2
INTERMEDIATE MATHEMATICS (STEM C) 44Example 3 Find the value of x if x x 3 2x 1 2 4 8 . Solution: x x 3 2x 1 2 4 8 x 3 2x 1 x 2 3 x 2x 6 6x 3 2 2 2 2 2 2 Therefore, Example 4 Solve for x. a) x 32 64 b) x 2 8 8 c) 2m 8 16 1 Solution: a) x 32 64 x 5 6 5x 6 2 2 2 2 5x 6 6 x 5 b) x 2 8 8 x 2 3 3 3 3x 6 2 2 2 2 2 3 3x 6 2 15 3x 2 5 x 2 x 2x 6 6x 3 3x 6x 3 3x 3 x 1