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  7 inch2.5 inch2.5 inch2.5 inch LO RITHM  I N X Topic Page No. LOGARITHM l. Basic Mathematics 12. Historical Development of Number System 33. Logarithm 54. Principal Properties of Logarithm 7 5. Basic Changing theorem 86. Logarithmic equations 107. Common & Natural Logarithm 128. Characteristic Mantissa 129. Absolute value Function 1410. Solved examples 17 11. Exercise 2412. Answer Key  3013. Hints & Solutions 31 LO RITHM  BANSALCLASSES Private Ltd. ‘Gaurav Tower’, A-10, Road No.-1, I.P.I.A., Kota-05  ACC MT LOGARITHM 1 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) isdividedby (x–a), thentheremainderisequalto 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) isafactorof p(x).Conversely,if (x–a)isafactorof p(x), then p(a)=0. Note :  Let p(x) beanypolynomialofdegreegreaterthanorequaltoone.Ifleadingcoefficientof p(x) is 1then p(x) iscalledmonic.(Leadingcoefficientmeanscoefficientofhighestpower.) SOME IMPORTANT IDENTITIES : (1) (a + b) 2 = a 2 + 2ab + b 2 = (a – b) 2 + 4ab(2) (a – b) 2 = a 2  – 2ab + b 2 = (a + b) 2  – 4ab(3) a 2  – b 2 = (a + b) (a – b)(4) (a + b) 3 = a 3 + b 3 + 3ab (a + b)(5) (a – b) 3 = a 3  – b 3  – 3ab (a – b)(6) a 3 + b 3 = (a + b) 3  – 3ab (a + b) = (a + b) (a 2 + b 2  – ab)(7) a 3  – b 3 = (a – b) 3 + 3ab (a – b) = (a – b) (a 2 + b 2 + ab)(8) (a + b + c) 2 = a 2 + b 2 + c 2 + 2 (ab + bc + ca) = a 2 + b 2 + c 2 + 2abc       c1 b1a1.(9) a 2 + b 2 + c 2  – ab – bc – ca =21   222 )ac()c b() ba(   (10) a 3 + b 3 + c 3  – 3abc = (a + b + c) (a 2 + b 2 + c 2  – ab – bc – ca)=21(a + b + c)    222 )ac()c b() ba(   If (a + b + c) = 0, then a 3 + b 3 + c 2 = 3abc.(11) a 4  – b 4 = (a 2 + b 2 ) (a 2  – b 2 ) = (a 2 + b 2 ) (a – b) (a + b)(12) If a, b    0 then (a – b) =     ba ba   (13) a 4 + a 2 + 1 = (a 4 + 2a 2 + 1) – a 2 = (a 2 + 1) 2  – a 2 = (a 2 + a + 1) (a 2  – a + 1)  2 ACC MT LOGARITHMBANSALCLASSES Private Ltd. ‘Gaurav Tower’, A-10, Road No.-1, I.P.I.A., Kota-05  Definition of Indices : Theproductof m factorseachequalto a isrepresented by a m .So, a m =a·a·a........a(mtimes).Here a iscalledthebaseand m istheindex (orpowerorexponent). Law of Indices : (1) a m+ n =a m ·a n ,where m and narerational numbers.(2) a  –m =  m a1 , provided a    0.(3) a 0 = 1, provided a    0.(4) a m– n = nm aa ,where m and narerational numbers, a   0.(5) (a m ) n = a mn .(6)  q pq p aa   (7) (ab) n = a n  b n . Intervals : Intervalsarebasicallysubsetsof R(thesetofallrealnumbers)andarecommonlyusedinsolvinginequaltities.If a,b  Rsuchthat a<b,thenwecandefinedfourtypesofintervalsasfollows: Name Representation Discription. Open interval (a, b) {x:a<x<b}i.e.,endpointsarenotincluded.Closeinterval [a,b] {x:a  x   b}i.e.,endpointsarealsoincluded.Thisispossibleonlywhenbothaandbarefinite.Open-closedinterval (a,b] {x:a<x   b}i.e.,aisexcludedand bisincluded.Closed-openinterval [a,b) {x:a  x<b}i.e.,a isincludedand bisexcluded. Note :(1) Theinfiniteintervalsaredefinedasfollows: (i) (a,   ) = {x : x > a } (ii) [a,   ) = {x |x    a }(iii) ( –    , b) = {x : x < b} (iv) (–    , b] = {x : x    b}(v) (–    ,   ) = {x : x   R} (2)  x  {1,2}denotessomeparticularvaluesof x,i.e., x=1,2. (3)  Iftheirisnovalueofx,thenwesay x   (i.e.,nullsetorvoidsetoremptyset).
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