Indices and LogarithmsChapter 2 Indices and LogarithmsIndicesIf a positive integer a is multiplied by itself three times we get a3 i.e. a a a a 3 . Here a is called the base and 3 the index or power. Thus a 4 means the 4th power of a. In general a n means the nth power of a where n is any positive index of the positive integer a.Rules of IndicesThere are several important rules to remember when dealing with indices. If a b m and n are positive integers then 1 2am an am n am an am na m n a m n a m b m a b me.g.35 38 313 514 53 51152 6 512 35 25 65e.g.e.g. e.g.345a a b bm mme.g.5 5 3 34 446 7a0 1an 1 anm ane.g.1 an e.g.50 15 3 1 832 831 538 9ane.g.m n m38 3 82 3 82 a ane.g.2.1Indices and Logarithms Example 2-1 Evaluate Solution i23ii1 83iii3 16 4iv253 2i2311 3 8 2ii1 833823 2iii3 4 16 1643iv2513 25 2 2381 253 1 1 3 5 125Example 2-2Simplify i1 a3 2 a5 1 a2iia b 324iii3a a521 1 2 a Solution1 2 1ia3 a5 a2iia 3b 2 4iii3a 5 a 2 a 1 21 a31 a31121 a3 5 27 30a34 24b2 5 a2 a5aa1 21 2aa b12 8a a1 2 1 3 5 217 30Solving Exponential EquationsExample 2-3 Solve the following exponential equations i ii 2 x 32 4 x 1 0.25Solution i 2 x 32 2 x 25 x 52.2Indices and Logarithms ii4 x 1 0.25 1 4x 1 4 x 1 4 4 1 x 1 1 x 2Example 2-4 Solve the equation 22 x 3 2 x 3 1 2 x . Solution 22 x 3 2 x 3 1 2 x 2 x 2 x 23 2 x 23 1 2 x Let y 2 x8 y2 8 y 1 y8 y2 7 y 1 0 8 y 1 y 1 0 1 y or 1 8 1 When y 8 1 2x 8 x 2 2 3 x 3Express in general form ax 2 bx c 0 . The quadratic formulae can be used here.when y 12 x 1 inadmissibleLogarithmsFor any number c such that a bc a 0 and a 1 the logarithm of a to the base b is defined to be c and is denoted by logb a . Thus if a bc then logb a c For example 81 34 log 3 81 42100 10 log10 100 2Note The logarithm of 1 to any base is 0 i.e. log a 1 0 . The logarithm of a number to a base of the same number is 1 i.e. log a a 1 . The logarithm of a negative number is undefined.2.3Indices and LogarithmsExample 2-5 Find the value ofi iiilog 2 64 1 log 3 9ii ivlog9 3log8 0.25Solution i Let log 2 64 xiiLet log 9 3 x64 2 x 26 2 x x 63 9x 3 32 x 1 2x 1 x 2iv Let log8 0.25 x 0.25 8 x 1 23 x 4 2 2 23 x 3x 2 2 x 3iiiLet log 31 x 9 1 3x 932 3x x 2Laws of Logarithms1Product Rule log a mn log a m log a n Quotient Rule m log a log a m log a n n Power Rule log a m p p log a me.g.log3 5 log 3 2 log 3 102e.g.5 log3 5 log 3 4 log 3 43e.g.log10 52 2 log10 52.4Indices and Logarithms Example 2-6 Without using tables evaluate log1041 41 log10 70 log10 2 log10 5 . 35 2Solution 41 41 log10 log10 70 log10 2 log10 5 35 2 41 2 41 log10 70 5 2 35 log10 100 log10 102 2 log10 10 2Changing the Base of LogarithmsLogarithms to base 10 such as log10 5 and log10 x 1 are called common logarithms. An alternative form of writing log10 5 is lg 5 .Common logarithms can be evaluated using a scientific calculator. Logarithms to base e such as log e 3 and log e x are called Natural logarithms or Napierian logarithms. Natural logarithms are usually written in an alternative form for example log e 3 is written as ln 3 . Note e 2.718... If a b and c are positive numbers and a 1 then log a b Example 2-7 Find the value of log5 16 .log c b . log c aSolution Since scientific calculators use the logarithm with base 10 we will have to change the base of log5 16 to a logarithm with base 10. log5 16 log10 16 1.204 1.722 log10 5 0.6992.5Indices and LogarithmsSolving Logarithmic EquationsExample 2-8 Solve the equation 3x 18 . Solution 3x 18 Taking logarithms to base 10 on both sides log10 3x log10 18 Using Power Rule x log10 3 log10 18 log10 18 1.2553 x log10 3 0.4771 2.631 Example 2-9 Given that log10 4 2 log10 p 2 calculate the value of p without using tables or calculators.Solution log10 4 2 log10 p 2log10 4 p 2 2Use the Power Rule and then the Product Rule24 p 10 100 p2 4 2 p 25 p 52Since p cannot be 5 because log10 5 is not defined p 5 .Example 2-10 Solve the equation log10 3x 2 2 log10 x 1 log10 5 x 3 .Solution log10 3x 2 2 log10 x 1 log10 5 x 3 log10 3 x 2 log10 x 2 log10 5 x 3 1 3x 25 x 3 log10 1 x2 3 x 25 x 3 101 2 x 2 15 x 9 x 10 x 6 10 x 2Use the Product Rule and Quotient Rule Convert to index form2.6Indices and Logarithms5x2 x 6 0 5 x 6 x 1 0x Since x cannot be negative x 1 .Express in the general form Quadratic formulae can be used here.6 or x 1 5Exercise1. Simplify each of the following giving your answer in index form1 1 1a da2 a3 a61 a 3 2 5 15 bba3 a 4 a 23c12a 4 4a 6ea af ab 2 2 4 9a b 32.Solve the following equations b a 3x 81 1 d 2x e 8 h g 5x 1 1 j 5 3 x 25 x 1 1255 x 125 1 16 x 2 4x 3 27 x 3c f i32 x 8 1 7x 49 x 4 32 x 63.By using an appropriate substitution or otherwise solve the following equations a c22 x 2 x 2 12bd22 x 1 92 x 2 1 010 3 x 1 x 9 283 3 0 32t1 3t24.Solve the simultaneous equation52 x y 625 5. Write the following in logarithmic form a b 52 25 120 1Write the following in index form a log 2 8 3 b24 x 2y 1 16c73 3436.log5 625 4clog 61 2 362.7Indices and Logarithms 7. Solve the following equations a log x 81 3 blog 2 25 xclog5 x 38.Simplify the following logarithms a 3log 6 5 log 6 25 c 2 log 3 5 log 3 10 3log 3 4blog 2 21 log 2 3 log 2 59.If log3 2 0.6309 and log 3 5 1.465 evaluate without the use of calculators a log3 10 b log3 15 c log3 2.5 d log3 12 Solve the following equations 3x 1 a log 2 3 2x 7 log3 x 2 5 x 9 1 cSolve the following equations a log10 x2 3log10 x10.blog3 2 x 1 log 3 x 7 2dlog5 3 x 2 20 x 50 211.blog 4 x2 5log 4 x 412.Solve the following equations giving your answers correct to 3 significant figures a b c 2x 5 3x 1 5 4x 1 7 d e f 3x 1 2 x 2 35 x 1 28 7 x 1 23 x 213.Solve the following equations for x giving your answers correct to 3 significant figures where necessary a 2 log 2 x 3 log 2 x 6 b log3 x log3 2 x 1 2 c52 x 165 x 632.8
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Chapter 02 - Indices and Logarithms
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