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Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ LECTURE 1 OF 7 TOPIC : 1.0 NUMBER SYSTEM SUBTOPIC : 1.1 Real Numbers LEARNING OUTCOMES : At the end of the lesson, students should be able to (a) define and understand natural numbers, whole numbers, integers, prime numbers, rational numbers and irrational numbers. (b) represent rational and irrational numbers in decimal form, (c) represent the relationship of number sets in a real numbers diagrammatically showing N W Z Q and Q Q = , where N, W , Z , Q , Q and R represent the set of natural, whole, integer, rational and real numbers respectively. SET INDUCTION: 1.1 Real Numbers The lecturer gives the situation such as; i. how many children in the family, ii. what their heights, iii. “Assume that a woman wants to buy fishes. The total price of the fishes is RM10, but the woman has only RM5. So she owes RM5 from the seller. (RM5 – RM10 = RM5). In conclusion, numbers can be classified into different types of numbers where dates of birth belong to natural numbers, the height is rational numbers and –RM5 is a negative number. CONTENT A) Bil. TYPES OF NUMBERS Type of numbers 1 Natural numbers 2 Prime numbers 3 Whole numbers Symbol N W Characteristics Example no zero N = {1, 2, 3 …} no negative numbers no decimal numbers Greater than 1 &; Only can be divided by itself = {2, 3, 5, 7 …} and 1 W = {0, 1, 2, 3 …}. ‘natural numbers’ + zero 1 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ Negatives numbers Z = {…, -3, -2, -1, 0, 1, 4 Integers Z Has zero 2, 3 …} No decimal numbers Positive whole numbers 5 Positive Z+ Z + = {1, 2, 3 …} Without zero Negative whole numbers Z ..., 3, 2, 1 6 Negative Z Without zero Numbers can be multiply or = {2k, k Z} 7 Even divide by 2. = {2, 4, 6, 8, …} = {2k + 1, k Z} 8 Odd Other than even numbers = {1, 3, 5, 7, …} a Can be written in quotient Q= ; a, b Z , b 0 . b @ ratio of two integers. 9 Rational numbers Q For terminating or repeating 3 5 4 = , , ,... decimals. 2 1 9 Numbers are not repeating or terminating. 10 Irrational numbers Q = 3, 10, , e Cannot be expressed as quotient /ratio. The real number consists of 11 The real number rational numbers and Q Q irrational numbers. B) THE DIFFERENCE BETWEEN RATIONAL NUMBERS AND IRRATIONAL NUMBERS. Rational Numbers Irrational Numbers 1 = 0.333… 3 2 = 1.41421356… = 3.14159265… 4 = 0.363636… 11 e 2.71828128... 1 = 0.25 4 2 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ C) RELATIONSHIP OF NUMBERS SETS [N, W , Z , Q , Q and R] i) Represented grammatically Q Q Z W N From the diagram, we can see that : 1. N W Z Q 2. Q Q = Example 1 For the set of {-5, -3, -1, 0, 3, 8}, identify the set of (a) natural numbers (b) whole numbers (d) even numbers (e) negative integers (c) prime numbers (f) odd numbers Example 2 1 2 , , 0, 4, 5.1212…}, identify the set of 3 (a) natural numbers (b) whole numbers (c) integers (d) rational numbers (e) irrational numbers (f) real numbers Given S = {-9, 7, Example 3 a b (b) 5.45959… Express each of this number as a quotient (a) 1.555… 3 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ LECTURE 2 OF 7 TOPIC : 1.0 NUMBER SYSTEM SUBTOPIC : 1.1 Real Numbers LEARNING OUTCOMES : At the end of the lesson, students should be able to (d) represent open, closed and half-open intervals and their representations on the number line. (e) simplify union, , and intersection, , of two or more intervals with the aid of number line. The Number Line The set of numbers that corresponds to all point on number lines is called the set of real number. The real numbers on the number line are ordered in increasing magnitude from the left to the right For example for –3.5, 4 2 and can be shown on a real number line as 3 3 2 3.5 Symbol a=b a<b a>b 1 0 1 2 3 Description a equal to b a less than b a greater than b Note: The symbols ‘<’ or ‘>’ are called inequality sign 4 2 3 4 Example 3=3 4 < 4 5>0 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ A) REPRESENTED ON NUMBER LINE. All sets of real numbers between a and b, where a b can be written in the form of intervals as shown in the following table. Type of Interval Notation Inequalities Closed interval [a, b] axb Open interval (a, b) a<x<b Half-open interval (a, b] Half-open interval Representation on the number line a b a b a<x b a b [a, b) a x<b a b Open interval (, b) < x < b b Half-open interval (, b] < x b b Open interval (a, ) a<x< a Half- open interval [a, ) a x< a Note : The symbol is not a numerical. When we write [a, ), we are simply referring to the interval starting at a and continuing indefinitely to the right. Example 4 List the number described and graph the numbers on a number line. (a) The whole number less than 4 (b) The integer between 3 and 9 (c) The integers greater than -3 Example 5 Represent the following interval on the real number line and state the type of the interval. (a) [-1, 4] (b) {x : 2 x 5} (c) [2, ) (d) {x : x 0, x R} 5 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ B) INTERSECTION AND UNION Intersection and union operation can be performed on intervals. For example; A = [1 , 6) and B = (2, 4), The Intersection of set A and set B is a half-open interval [1,4). The union of set A and set B is given by A B = (2, 6) is an opened interval. All these can be shown on a number line given below: AB AB B 2 A 1 4 6 Example 6 Solve the following using the number line (a) [0, 5) (4, 7) (c) (, 0] [0, ) (b) (, 5) (1, 9) (d) (4, 2) (0, 4] [2, 2) Example 7 Given A = {x : -2 x 5} and B = {x : 0 x 7}. Show that A B = (0, 5]. 6 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ EXERCISES 1: 1. Given {-7, - 3 , 0, (a) N 1 , , 5 2 4 , 0.16, 0.8181…}. List the numbers for (b) Z (c) Q (d) R 2. Express each of there numbers as a quotient (a) 0.444… (b) 0.5353… (e) Q a b (c) 1.777… 3. Represent these intervals on a number line. (a) [2, 5] (b) (4, 5] (c) (8, 8) (d) [2, ) 4. Write down the following solution set in an interval notation. (a) {x : x 6} (b) {x : 3 x < 6} (c) {x : 5 < x 5} (d) {x : 0 x 4} (e) {x : x 8} {x : x 13} (f) {x : x 4} 5. Given A = [2, 5], B = (3, 5], C = (7, 7) and D = [3, ). Find (a) A B (b) A B (c) A C D (d) C D A Answers: 1. (a) none (b) {-7, 0, (c) {-7, 0, 4} 1 , 5 4 , 0.16, 0.8181…} (d) {-7, - 3 , 4 , 0, (e) {- 3 , 2. (a) 4 9 } 2 (b) 53 99 1 , , 0.16, 0.8181…} 5 2 (c) 17 9 7 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ 3. (a) 2 5 (b) -4 5 (c) -8 8 (d) -2 4. (a) [6, ) (b) [-3, 6) (c) (-5, 5] (d) [0, 4] (e) (, 8][12, ); half- open interval (f) (-, 4] ; half-open interval 5. (a) [2, 5] (b) (-3, 5] (c) [2, 5] (d) [2, 5] 8 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ LECTURE 3 OF 7 TOPIC : 1.0 Number System SUBTOPIC : 1.2 Complex Numbers LEARNING OUTCOMES : At the end of the lesson, students should be able to (a) represent a complex number in Cartesian form. (b) define the equality of two complex numbers. (c) determine the conjugate of a complex numbers. (d) To perform algebraic operations on complex numbers. SET INDUCTION: Look at this equation: x2 + 1 = 0 This equation does not have real roots as we cannot find a value for x whenever x = 1 .This problem was encountered by Heron Alexandra. One hundred years later, Mahavira from India stated that a negative value does not have square root because there is no number that is squared to produce it. In 1637, Descrates of France, introduced ‘real number’ and ‘imaginary number’. This idea was used by Euler from Switzerland who defined it as 1 in 1948. However ‘complex number’ was introduced after one hundred years later by Gauss from Germany (1832). CONTENT We have already met equations such as x2 = -1 whose roots are clearly not real since when squared they give –1 as the result. The imaginary number I, defined to be solution to the equation x2 = -1, is a basis of this new set. The imaginary unit i Defined as i = 1 , where i2 = -1 Using the imaginary unit i, we can express the square root of any negative number as a real multiple of i. For example, 16 = 1 16 = i 16 = 4i, 3= i 3 9 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ Power of i can be simplified therefore 1 = -1 = 1 1 = -i = 1 1 = 1 1 =1 2 i2 = i3 i4 2 1 2 2 Complex Numbers, Definition: The set of all numbers in the Cartesian form, z = a + bi with real numbers a and b, and i the imaginary unit, is called the set of complex numbers. The real number a is called the real part, and the real number b is called the imaginary part, of the complex numbers. If a = 0 and b ≠ 0, then the complex number bi is called a pure imaginary number. Cartesion form: z = a + bi b= Imaginary part of complex numbers a = real part a & b = real numbers = Re (z) Example 8 (a) If z = 2 + 3i , (b) If z = 4 – 2i , + i = imaginary unit = Im (z) = Set of complex numbers Re(z) = 2 , Im (z) = 3 Re (z) = 4 and Im(z) = -2 If b = 0 z = a + i(0) = a so it is a real number. Real numbers, then, are a subset of complex numbers, C . a) Equality of Complex Numbers. For two complex numbers z 1 = a + bi and z 2 = c + di, z 1 = z 2 if and only a = c and b = d. Example 9 Find the value of m and n if z 1 = z 2 for z 1 = 3 + 2i and z 2 = m + ni. Example 10 Find a and b for the following equations (a) a + b + (a – b)i = 6 + 4i (b) a + 2b + (a – b)i = 9 + 0i 10 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ b) Conjugate of Complex Numbers The complex conjugate of a complex number, a + bi is a – bi, and the conjugate of a – bi is a + bi. The multiplication of complex conjugates gives a real number. (a + bi)(a - bi) = a2 + b2 (a – bi)(a + bi) = a2 + b2 Complex Numbers, Conjugate of a + bi a - bi a - bi a + bi **This fact is used in simplifying expressions where the denominator of a quotient is complex. Example 11 Simplify the expressions: a) 1i 3 1+ i 4 + 7i c) 2 + 5i 2i d) 3i 2 b) c) Adding and subtracting complex numbers Complex numbers can add together by adding the real parts and then adding the imaginary parts. You can subtract one complex number from another by subtracting the real parts and then subtracting the imaginary parts. So : (a + bi) + (c + di) = (a + c) + (b + d)i ( a + bi) – (c + di) = (a – c) + (b – d)i Example 12 Given that z = 2 + 3i and w = 7- 6i , find (a) z+w (b) w–z 11 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ d) Multiplying one complex number by another To multiply two complex numbers together, apply the rules of algebra. So : (a + bi) (c + di) = ac + adi +bci +bdi2 = ac + (ad + bc)i – bd ( i2 = -1 ) = (ac – bd) + (ad + bc)i Example 13 Given that (a) z = 4 + 3i and w = 7 +5i , find zw. (b) w = 4 – 2i and z = -7 + 5i, find zw. e) Solving equations Just as you can have equations with real numbers, you can have equations with complex numbers, as illustrated in the example below. Example 14 Solve each of the following equations for the complex number z. 4+5i = z - (1-i) Example 15 Solve (x + yi)(3 – i) = 1 + 2i where x and y are real. Example 16 1 i If z = , find z in the Cartesian form a + bi. 2i Example 17 If z =1 – 2i, express z + 1 in the form a + bi z 12 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ EXERCISE 2: 1) Write the following complex numbers in the form x + yi a) (3 +2i) + ( 2 + 4i) b) (4 + 3i) - ( 2 + 5i) c) (3 + 2i)( 4 - 3i ) Ans: a)5 + 6i b) 2 – 2i c) 18 - i 2. Solve for z when a) z( 2 + i) = 3 – 2i b) (z + i)(1 – i) = 2 + 3i 1 1 3 c) + = z 2 i 1 i Ans : a) 4 7 i 5 5 b) 1 3 i 2 2 c) 11 17 i 10 10 3) Find the values of the real numbers x and y in each of the following: x y 1 a) + 1 i 1 2i x yi 2 b) 2 i i 3 1 i Ans : 4 5 ,y 3 3 4 18 b) x , y 5 5 a) x 13 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ LECTURE 4 OF 7 TOPIC : 1.0 NUMBER SYSTEM SUBTOPIC : 1.3 Indices, surds and Logarithms LEARNING : At the end of the lesson, student should be able to OUTCOMES (a) state the rules of indices SET INDUCTION: The words ‘square’ and ‘cube’ come from geometry. If a square has side x units, the square of the number x gives the area of the square. If a cube has side x units, the cube of the number x gives the volume of the cube. These are written as x2 and x3 .This notation can be extended to higher powers such as x4 and x5, although there is no obvious physical interpretation of these expressions. The index notation is used to save us having to write several multiplications. 56 means 5 5 5 5 5 5 As defined, the notation is restricted to positive whole numbers, but it can be further extended to 1 negative and fractional powers, so that we can speak of 2-3, or of 3 2 . CONTENT: We can write 2 2 2 2 2 as 25 .This is known as the power (sometimes called an exponent or index); a to the power x is written as ax .So ax is an expression in which a is the base and x is the power. INDICES Rules of indices 1. am x an = a(m+n) 2. (am ) n = a m x n = a mn am m n 3. a a = n = a m n a 4. a 0 = 1 provided a 0 1 5. a- m = m a power x a base 1 6. 7. 8. 9. am = m n m n a m 1 n a = a = (a ) m (ab ) n = an bn n an a n b b 14 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ There are some special cases to consider. 1. Division when powers are equals (Zero index) 63 63 = 63 - 3 1= 60 This result leads to another rule. a 0 = 1 provided a 0 2. Division when the power of the denominator is greater than that of the numerator (Negative index) 73 75 = 73 – 5 777 = 7-2 77777 1 = 7-2 72 a- m = 1 am A negative power indicates a reciprocal. 3. Rational index Rule 1 can be used to introduce rational power. 1 Suppose that p = q = in rule 1 2 1 1 1 2 1 a2 x a2=a2 1 2 (a ) 2 = a 1 = a 1 2 = a 1 The meaning of power can be established in a similar way. 3 a 1 3 1 3 a x a x a 1 3 (a ) 3 1 3 = a 1 1 1 3 3 3 = a1 = a 1 a3 = 3 a 15 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ 1 1 1 This means ‘power ’ implies a cube root. Investigate powers , , and so on in the same way. 3 4 5 Notice how the fraction is related to the root. a 1 m = m a Example 18 Simplify: 35 3 6 (a) 34 (b) 18 x 2 y 5 3x 4 y 3 p q r 2 (c) 4 3 Example 19 Write the following expressions in index form: a) 2 x3 c) x 4 2 1 1 2x 4 b) d) 3 54 x 4 2x Example 20 Without using calculator, evaluate: (a) 9 3 2 (b) 11 1 25 Example 21 Simplify the following expression 3 a 2 b 3 x 2 b 1 1 2 3 1 (a) x y a2 y3 1 2 0.04 2 3 (c) 2 2 n 4 24.2 2n1 (b) 10 2 n 2 EXERCISE 3: 1. Find the value of each of the following a) 64 2 3 1 27 3 b) 8 c) (0.04) 3 2 d) 2 n 8 3n 4 5 n Answer a) 16 b) 2 3 c) 125 d) 1 16 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ LECTURE 5 OF 7 TOPIC : 1.0 NUMBER SYSTEM SUBTOPIC : 1.3 Indices, surds and Logarithms LEARNING : At the end of the lesson, student should be able to OUTCOMES (a) Explain the meaning of a surd and its conjugate, and to carry out algebraic operations on surds. SET INDUCTION A calculator will give 2 as 1.414213562, to 10 significant figures. This is not the exact value, because 2 is irrational, and it can never be represented exactly as a fraction or as terminating decimal. A number expressed in terms of root sign is a surd or radical. There are many advantages in expressing the number in this way, instead of writing it as a decimal. CONTENT: Expression of Root sign Radical exp : Surd 4 2 exp : 27 3 3 #Answer equal to whole number Some examples of surds are 2 , approximation is precise. 7 5 and 71 #Answer equal to irrational number 7 . They cannot be evaluated exactly, i.e. no 17 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ A) RULES OF SURDS a × ab = a b b a = b , a, b 0 PROPERTY 1 , a, 0 , b 0 PROPERTY 2 a b + c b = (a + c) b PROPERTY 3 a b - c b = (a - c) b PROPERTY 4 B) ADDITIONAL RULES 1) a a a 2) a + 3) a 4) ( a + b ) 2 = a + b+ 2 ab 5) ( a + b )( a - a = 2 a b = a b b ) = a2 – b2 Remark: a b ( a + b ) b) 24 Example 22 Simplify: 45 a) c) 6 7 2 7 C) Multiplying Radicals There are 2 types. 1) a b = ab 2) a b c d = ac bd 18 d) 5 3 27 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ Example 23 Multiply: 1. 3 6 5 7 3. 7 8 10 6 3 (4 7 - 3 ) 4. 8 2 (5 6 + 5. (2 3 +4 2 )(6 3 +2 2 ) 2. 2) Example 24 Expand and simplify ( 8 3 ) ( 8 3 ) . The conjugate of number. a + For example, ( 8 + b is 3 )( 8 - a – b )( a – b where ( a + b ) = a – b a rational 3 ) = ( 8 )2 - ( 3 )2 = 8 – 3 = 5 D) Rationalising the denominator When the denominator of a fraction contains a square root, we generally simplify the expression by rationalising the denominator. To rationalize a denominator, multiply the numerator and the denominator of the fraction that will result in the denominator to become a rational number. If Denominator Contains the Factor 3 To Obtain Denominator Free from surds Multiply by 3 3 1 3 1 2 3 2 3 3 3 1 3 1 2 2 3 2 9 7 5 3 5 3 2 3 2 2 2 2 2 2 2 5 3 5 3 In rationalizing the denominator of a quotient, be sure to multiply both the numerator and the denominator by the same expression. 19 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ Example 25 Rationalise: 2 3 3 5 1 (a) (b) (c) (d) 5 3 2 3 3 7 2 Example 26 Rationalise the denominator and simplify. 1 2 1 3 17 5 (a) (b) 1 2 1 2 17 5 (c) E) Solving Surd Equation Example 27 Solve each of the following equation : 2x 1 5 0 (a) (c) 3x 1 + 1 = x x + x2 =2 (d) x 13 7 x 2 (e) 3x 1 2 x 1 x 2 (b) EXERCISE 4: 1. Simplify 63 3 75 2 48 5 12 (a) (b) Answer : 2. (a) 7 (b) 3 3 Expand and simplify: (a) (2 - 3 3 )(3 + 2 3 ) (b) (5 - 2 7 )(5 + 2 7 ) Answer : (a) -12 - 5 3 (b) -3 20 3 2 3 2 2 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ LECTURE 6 OF 7 TOPIC : 1.0 NUMBER SYSTEM SUBTOPIC : 1.3 Indices, surds and Logarithms LEARNING : At the end of the lesson, student should be able to OUTCOMES (a) state and prove the laws of logarithm such as loga MN = loga M + M loga N, loga = loga M - loga N and loga MN = N loga M. N SET INDUCTION: The Richter scale measures the magnitude of earthquakes using logarithms. The Richter value is only part of the number describing on earthquake is 106. Many people believe an earthquake with a Richter value of 8 is twice as strong as one with value of 4. How do the two actually compare in intensity? CONTENT A) Common Logarithms Logarithm is a number y (y > 0) for any given base a (a > 0) and is written as log a y x where y= a x and x R. B) Natural Logarithms The natural logarithmic function and the natural exponential function are inverse functions of each other. The symbol ln x is an abbreviation for loge x, and we refer to it as the natural logarithms of x. Definition of Common Logarithms log x = log10x , for every x > 0 Definition of Natural Logarithms ln x = loge x , for every x > 0 21 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ Example 28 1. 2 3 = 8 thus 3 is the power to which the base 2 must be raised to obtain 8 or 3 is the logarithm which, with a base 2, gives 8. This is written simply as 3 = log 2 8 2. 3. 32 = 9 thus 2 is logarithm which, with a base 3, gives 9 or 2 = log 3 9 1 ( ) 2 = 25 5 1 , gives 25 or -2 = log 1 25 5 5 So, we see that the base of a logarithm may positive be any number. Common logarithms have base 10. From a calculator it is found that the logarithm of 5 is 0.6990 thus -2 is logarithm which, with a base therefore 10 0.6990 = 5 or log 10 5 = 0.6990 It is usual to omit the base 10 and to write simply lg 5 = 0.6990 or log 5 = 0.6990 But any base other than 10 must be stated. Therefore log 100 = 2 (i.e 100 = 10 2 ) And log 0.1 100 = -2 (i.e 100 = 0.1 2 ) In general, log a b = c b = a c Example 29 For each of the following, write down an expression for a logarithm in a suitable base: 1 1 (a) 100 = 10 2 (b) = 5 32 2 Example 30 Without using calculator, find the value of: 1 log 1 (4) (a) log2 (b) 8 2 22 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ (C) The Laws of Logarithms Let Therefore log a b = x ax = b and log a c = y and ay = c a x . a y = bc or a x y therefore x + y = log a bc i.e Similarly = bc log a b + log a c = log a bc ax ay b c = b c Or a x y therefore x - y = log a b c log a b - log a c = log a b c i.e Let log a b n RULE 1 = RULE 2 = z Therefore az So an = bn z Therefore log a b =b = z n Or n log a b = z i.e n log a b = log a b n RULE 3 23 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ The rule 1,2 and 3 are known as the three basic laws of logarithms, i.e. 1. Product rule : log a MN = log a M + log a N 2. The Quotient rule: log a M = log a M - log a N N 3. The power rule: log a M N = N log a M The following table lists the general properties for natural logarithmic form. Logarithms with base a Common logarithms Natural logarithms 1) loga(1) = 0 log 1 = 0 ln 1 = 0 2) logaa = 1 log 10 = 1 ln e = 1 3) logaax = x log 10x = x ln ex = x 4) a loga ( x ) x 10 log(x ) x e ln(x ) x Example 31 Expand: (a) ln 4x e3 ln (c) 7 (b) log 10x (d) ln x Example 32 Given log 2 = 0.301 and log 6 = 0.778 find log 12. Example 33 Given log 6 = 0.778, find log 36. Example 34 Write the following as single logarithms: a) log 8 – log 6 + log 9 2 ln (x + 7) – ln x b) 24 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ (D) Properties of Logarithms The properties of logarithms are very similar to the properties of exponents because logarithms are exponents. The properties will be used in solving logarithmic equations. Properties of logarithms Example 1) log a a 1 log5(5) = 1 2) loga(1) = 0 log2(1) = 0 3) loga(am) = m log (107) = 7 4) a loga ( m ) m 5 log5 ( 3) 3 1 5) log a log a ( N ) N 1 log log 1 log 3 3 log 3 6) loga(m) = loga(n) m=n log x = log 4 x = 4 25 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ LECTURE 7 OF 7 TOPIC : 1.0 NUMBER SYSTEM SUBTOPIC : 1.3 Indices, surds and Logarithms LEARNING : At the end of the lesson, student should be able to OUTCOMES (a) change the base of logarithm using, loga M = log b M . log b a (E) Change of Base Tables are not readily available which list the values of expressions such as log 7 2. However if log 7 2 = x 7x = 2 x log 7 = log 2 log 2 Or x = log 7 Therefore we have changed the base from 7 to 10 and can now use our ordinary tables or calculator to evaluate log 7 2. Then So Therefore log 7 2 = 0.3010 log 2 = = 0.3562 0.8451 log 7 A general formula for changing from base a to base b can be derived in the same way. If log a c = x Then So c = ax log b c = x log b a x= log b c log b a or log a c log b c log b a in the special case when c = b , this identity becomes log a b log b b log b a 26 Mathematics DM015 Topic 1 : Number System ______________________________________________________________________________ or 1 log a b log b a Example 35 Find the following expression to four decimal places. (a) log35 (b) log510 Example 36 Find x if ln x = 4.7 Example 37 Change to logarithmic form for the expression e7 = p EXERCISE 5: 1. Without using calculator, find the values of: (a) log 1000 (b) log 0.01 2. Find the values of the following, to three significant figures: 1 (a) log 50 (b) log( ) 3 3. Without using calculator, find the exact values of x in: (a) 4. x = log 2 64 (b) x = log 3 ( 1 ) 81 Find the values of the a (a) 3 = log a 125 (b) -2 = log a ( 27 1 )(c) 25 1 = log a 25 2