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# Note for Logarithm - L by Placement Factory

• Logarithm - L
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I N D E X Topic Page No. LOGARITHM l. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Basic Mathematics Historical Development of Number System Logarithm Principal Properties of Logarithm Basic Changing theorem Logarithmic equations Common & Natural Logarithm Characteristic Mantissa Absolute value Function Solved examples Exercise Answer Key 13. Hints & Solutions L O GA R I T H M 1 3 5 7 8 10 12 12 14 17 24 30 31

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1 ACC-MT- LOGARITHM LOGARITHM BASIC MATHEMATICS : Remainder Theorem : Let p(x) be any polynomial of degree geater than or equal to one and 'a' be any real number. If p(x) is divided by (x – a), then the remainder is equal to p(a). Factor Theorem : Let p(x) be a polynomial of degree greater than or equal to 1 and 'a' be a real number such that p(a) = 0, then (x – a) is a factor of p(x). Conversely, if (x – a) is a factor of p(x), then p(a) = 0. Note : Let p(x) be any polynomial of degree greater than or equal to one. If leading coefficient of p(x) is 1 then p(x) is called monic. (Leading coefficient means coefficient of highest power.) SOME IMPORTANT IDENTITIES : (1) (a + b) 2 = a 2 + 2ab + b2 = (a – b)2 + 4ab (2) (a – b)2 = a2 – 2ab + b2 = (a + b)2 – 4ab (3) a2 – b2 = (a + b) (a – b) (4) (a + b)3 = a3 + b3 + 3ab (a + b) (5) (a – b)3 = a3 – b3 – 3ab (a – b) (6) a3 + b3 = (a + b)3 – 3ab (a + b) = (a + b) (a2 + b2 – ab) (7) a3 – b3 = (a – b)3 + 3ab (a – b) = (a – b) (a2 + b2 + ab) (8) 1 1 1 (a + b + c)2 = a2 + b2 + c2 + 2 (ab + bc + ca) = a2 + b2 + c2 + 2abc     . a b c (9) a2 + b2 + c2 – ab – bc – ca = (10) a3 + b3 + c3 – 3abc = (a + b + c) (a2 + b2 + c2 – ab – bc – ca) =  1 2 2 2 2 ( a  b )  ( b  c)  ( c  a )   1 (a + b + c) (a  b) 2  (b  c) 2  (c  a ) 2 2  If (a + b + c) = 0, then a3 + b3 + c2 = 3abc. (11) a4 – b4 = (a2 + b2) (a2 – b2) = (a2 + b2) (a – b) (a + b) (12) If a, b  0 then (a – b) = (13) a4 + a2 + 1 = (a4 + 2a2 + 1) – a2 = (a2 + 1)2 – a2 = (a2 + a + 1) (a2 – a + 1)  a b  a b 

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2 ACC-MT- LOGARITHM Definition of Indices : The product of m factors each equal to a is represented by am . So, am = a · a · a ........ a ( m times). Here a is called the base and m is the index (or power or exponent). Law of Indices : (1) am + n = am · an, where m and n are rational numbers. (2) a–m = (3) 1 , provided a  0. am a0 = 1, provided a  0. (4) am – n = (5) am , where m and n are rational numbers, a  0. an (am)n = amn. p q (6) a  ap (7) (ab)n = an bn. q Intervals : Intervals are basically subsets of R (the set of all real numbers) and are commonly used in solving inequaltities. If a , b  R such that a < b, then we can defined four types of intervals as follows : Name Open interval Representation (a, b) Discription. {x : a < x < b} i.e., end points are not included. Close interval [a, b] {x : a  x  b} i.e., end points are also included. This is possible only when both a and b are finite. Open-closed interval (a, b] {x : a < x  b} i.e., a is excluded and b is included. Closed-open interval [a, b) {x : a  x < b} i.e., a is included and b is excluded. Note : (1) The infinite intervals are defined as follows : (i) (a, ) = {x : x > a } (ii) (iii) ( – , b) = {x : x < b} (iv) (v) (– , ) = {x : x  R} [a, ) = {x | x  a } (– , b] = {x : x  b} (2) x  {1, 2} denotes some particular values of x, i.e., x = 1, 2. (3) If their is no value of x, then we say x (i.e., null set or void set or empty set).

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3 ACC-MT- LOGARITHM Proportion : When two ratios are equal, then the four quantities compositing then are said to be proportional. If a c  , then it is written as a : b = c : d or a : b : : c : d. b d Note : (1) (2) a and d are known as extremes while b and c are known as means. Product of extremes = product of means. (3) If a c b d    (Invertando) b d a c (4) If a c a b    (Alternando) b d c d (5) If a c a c a b cd   1 = 1  = (Componendo) b d b d b d (6) If a c a c a b cd   1 = 1  = (Dividendo). b d b d b d (7) If a c a b cd   = (Componendo and dividendo) b d a b cd (8) If a b  then b2 = ac. Here b is called mean proportional of a and c. b c Histrorical Development of Number System : I. Natural Number’s Number’s used for counting are called as Natural number’s. {1, 2, 3, 4, ...........} II. Whole number’s Including zero (0) | cypher | ’kwU; | duck |love| knot along with natural numbers called as whole numbers. w = {0,1, 2, 3 ..............} i.e. N W 0 is neither positive nor negative III Integer’s Integer’s given by I = {...........–2, –1, 0, 1, 2, 3.......} i.e. N W I Type of Integer’s (a) None negative integers (b) Negative integers (I–) (c) Non positive integers (d) Positve integers (I+) { 0, 1, 2, 3, .........} {............–3, –2, –1} {............–3, –2 –1, 0} {1, 2, 3 ..........}