SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms
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Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18 Significnt Figures.20 3
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INDICES Index nottion Consider the following: 3 4 3 3 3 3 81 5 3 5 5 5 125 2 7 2 2 2 2 2 2 2 128 In generl n n fctors The letter n in n is clled one of three things: n is the index in n with s the bse n is the exponent or power to which the bse is rised n is the logrithm, with s the bse When number such s 125 is written in the form 5 3 we sy it is in exponentil or index nottion. First Index Lw Expressions in index form cn be multiplied or divided only if they hve the sme bse. To multiply index expressions dd the indices. 2 3 2 2 2. 2. 2 2. 2 2 5 [ 5 fctors of 2 multiplied together] Therefore 2 3 2 2 2 5 In generl: m n m+n Second Index Lw To divide index expressions subtrct the indices 3 5 3 3 3 2 Therefore 3 5 3 3 3 5 3 3 2 [Cncelling three lots of 3] In generl: m n m n, 0 Third Index Lw When n expression in index form is rised to power, multiply the indices. ( ) mens 5 2 5 2 5 2 From the First Index Lw 5 2 5 2 5 2 5 2 + 2 + 2 5 6 Therefore ( ) 5 6 In generl: ( m ) n m n nd ( m b p ) n mn b pn 5
Exmples 1) x 5 x 6 x 11 [ using first index lw] 2) y 8 y 3 y 5 [using second index lw] 3) (2 ) 2 3 x 6 8x 6 [using third index lw] Zero Index So fr we hve only considered expressions in which the indices re positive whole numbers. The index lws lso pply when the index is zero or negtive. Any expression with zero index is equl to 1. Consider 2 3 2 3 : Using the second index lw this is 2 3 2 3 2 3 3 2 0 But 2 3 2 3 1 So 2 3 2 3 2 0 AND 2 3 2 3 1 Therefore 2 0 1 In generl: 0 1, 0 Exmples 1) 7 0 1 2) (xy) 0 1 3) ( ) 1 4) (28x2 ) 0 1 Negtive Indices Consider 1 2 4 Therefore 2-4 2 0 2 4 [becuse 2 0 1] 2-4 [using the second index lw] In generl: -n n nd n n, 0 Exmples Express the following with positive indices: 1) 2-3 2) x -1 3) 2y -1 4) 5) ( ) (-2) 3 6) 2 x 2 y 3 7) ( ) 6
Combining index lws Index lws re used to simplify complex expressions. Exmples 1. Simplify (4 2 b) 3 b 2 (4 2 b) 3 b 2 4 3 6 b [using combintion of the second nd third lws] 2. Simplify ( ) ( ) ( ) ( ) [remove brckets] [convert division to multipliction] [express with positive indices] 3 2 3 6 + 3 b 2 + 3 c 6 4 [pply index lws] 3 1 9 b 5 c 2 3. Write x -1 + x 2 s single frction x -1 + x 2 + x 2 [express with positive indices] [dd using common denomintor, x] Exercise 1.Simplify the following ) c 5 c 3 c 7 b) 3 2 2 2 3 c) 3 2 b 3 b 4 d) 3 6 3 4 e) 8 3 f) x 4 y 6 x 2 y 3 g) (x ) h) (x m y n ) 2.Write with positive indices nd evlute if possible: ) x -5 b) 250 0 c) 3b -5 d) (pq) -2 e) (5xy) -3 f) g) 2-5 h) (-2) -3 i) -(3-2 ) j) 2 (-5) -2 3.Simplify these expressions giving your nswer in positive index form: ) 2 3 b 2-1 b 3 b) (5x -2 y) -3 c) d) x(x x -1 ) e) f) ( ) ( ) Answers 1. ) c 15 b) 96 c) 6 b 7 d) 9 e) 5 f) x 2 y 3 g) x 12 h) x 5m y 5n 2. ) b)1 c) d) e) f) 2yz 5 g) h) - i) - j) () 2 2 b 5 b) c) 11 d) x 2 1 e) f) 7
FRACTIONAL INDICES Expressions of the form The index lws pply to frctionl indices s well s positive nd negtive integer indices. Using the first index lw we know tht 3 3 3 1 3 Tht is 3 multiplied by itself equls 3. The squre root of 3, 3 is lso number tht, when multiplied by itself, equls 3: 3 3 3 Since 3 behves like 3 we sy tht 3 3. Similrly 2 2 2 2 nd 2 2 2 2 2 using the first index lw Since 2 behves liked 2 we sy tht 2 2 In generl: n n (nth root of where n is positive integer) Exmples 1) 4 4 2 2) 2 2 3 3) 3 3 4) 5) 6) 32 In most cses the root of number will not be ble to be written s frction nd will be n irrtionl number. For exmple 2 1.414.. Expressions of the form If, wht mening cn be given to? cn be written ( )..using the third index lw So ( ) In generl: m n n m where m nd n re both integers Exmples 1) 2) 3) 4) 5) 8
The index lws cn be used to evlute nd simplify expressions with frctionl indices. Exmples Simplify the following, nd evlute if possible. 1) ( ) [using the third index lw] 2) 3 3 3 3-1 [using second index lw] 3) 32 (2 ) 2 3 8 [write 32 s 2 5 ] 4) ( ) Exercise 1. Evlute the following expressions. If the nswer is not exct give the deciml pproximtion to two deciml plces. () 64 (b) 12 (c) 81 (d) 2 0 2. Simplify the following expressions. Give your nswer in index nottion with positive indices. () ( ) (b) (c) (12 ) (d) ( ) ( ) Answers 1. () 8 (b) 5 (c) (d) 3.02 2. () (b) (c) (d) 9
Logrithms The logrithm of number is the power to which the bse must be rised to give the number. This mens tht logrithm is n index. Now, let s revise wht it mens. 8 2 3 Index or Number bse This fmilir sttement my be written s log 2 8 3 Generlly: x n log n x (follow the rrows nd red x n, >0) Exmples Re-write these equtions in logrithmic nottion: () 5 3 125 log5125 3 Using the bove generl expression: 1 3 2 (b) 9 1 log3 2 9 Using the bove generl expression (c) 1 2 25 5 1 log25 5 2 bse Using the bove generl expression Index or Exmples To evlute logrithm: write the number in index form, with sme bse s the logrithm. use the definition of logrithm to evlute. Evlute log 5125 125 5 3 125 in index form with bse 5 log5125 3 equivlent logrithm sttement log Evlute 0.0001 0.0001-4 0.0001 in index form with bse log0.0001 4 equivlent logrithm sttement Two importnt logs to remember 0 1 0 1 log 10 In words: For ny bse, the log of 1 is zero. 1. 1 log 1 In words: The log of ny number with itself s bse is 1.
Logrithm Lws Corresponding to the three index lws, there re three lws of logrithms to help in mnipulting logrithms. First Logrithm Lw log mn log m log n Second Logrithm Lw m log log m log n Third Logrithm Lw p log m p log Exmples m log 15 log n 53 log 5 log 3 Using the first logrithm lw 12 log log 12 log 5 5 Using the second logrithm lw 2 log 9 2log 9 2log 4log 3 3 2 Using the third logrithm lw Note: To use the logrithm lws, ll logrithms must hve the sme bse. These lws together with the definition of logrithm cn be used to simplify nd evlute logrithmic expressions nd to solve equtions involving logrithms. Exmples 1.Solve log 2 x 5 log 2 x 5 2 5 x using the definition of logrithm Therefore x 32 2.Simplify log 6 log 2 log 6 log 2 log 6 2 log 12 using first log. lw log 3. Simplify 36 log3 b log3 2 log 6b log 2 3 3 6b log3 2 log 3b 3 Use the lws for dding nd subtrcting logrithms. 11
4. Simplify 1 log 36 log 15 2log 5 2 1 2 log 36 log 15 log 5 log 6 log 15 log 25 6 25 log 15 log 1 2 NB: 1 1 log 36 2 2 2 log 5 log 36 nd must be written s 2 log 5 nd before using ddition nd subtrction lws. [ use lws for dding nd subtrcting logs] 5. Solve for x log 2x 1 log Therefore 2x 1 3 3 x 1 both sides of the eqution re log so cn equte (2x+1) nd 3 Exercises Write in logrithm form. 2 4 () 3 9 (b) 000 2 (c) 0.01 (d) e b Exercise 2 Evlute without using clcultor. log () 7 49 (b) log log (c) 51 (d) log0 000 (e) log5 5 Exercise 3 1. Simplify. log 8log 3log 2 () 4 4 4 3 log 2 2log 3 log 18 (c) e e e e 1 log 3log b log b (e) 2 2.Solve for x log log 4 log 2 () x 2 2 1 log 25 log 4 2log 3 (b) 2 log 4 2log 3 2log 6 (d) log 2x 2log 3 log 6 (b) 2 2 2 Answers log 9 2 log (b),000 4 log (c) 0.01 2 () 3 Exercise 2 () 2 (b) 1/2 (c) 0 (d) 5 (e) 1 Exercise 3 log 12 log 1.() 4 (b) 2. () x 2 (b) x 27 45 log b 4 (c) 3 (d) 0 (e) log b (d) e 12
EXPONENTIAL EQUATIONS Exponentil equtions hve term in which the vrible is power ( index or exponent). For exmple: x b nd 2 1-x 4 3x If two equl numbers cn be expressed with the sme bse then the indices must be equl: ie. m x m y x y [equting indices] We cn use this property to solve exponentil equtions. Exmples 1. 3 x 27 3 x 3 3 [27 is written in index form so tht both sides hve bse of 3]. x 3 [Equte indices] 2. 2 1-x 2 1-x 2-3 [ is written in index form with bse of 2] 1 x -3 [equte indices nd solve for x] x 4 3. 25. 5 2 [25 is written in index form, with bse of 5] 2 [Equte indices nd solve for x] x Not ll equtions hve integer solutions. For exmple: 3 x hs solution between 2 nd 3 since 3 2 9 nd 3 3 27 Logrithms with bse or bse e cn be used to solve equtions like this with the clcultor. On the clcultor use the log button to evlute logrithms to bse nd the ln button to evlute logrithms to bse e. Exmples 1. 3 x log (3 x ) log () x log (3) 1 [Tke logs (bse ) of both sides] [using log lws] x ( ) x x 2.095. [using clcultor] 13
2. 2 5 x + 1 15 5 x + 1 7.5 log (5 x + 1 ) log (7.5) [Tke logs of both sides] (x + 1)log (5) log (7.5) [using log lws] (x + 1) x + 1 1.252 x 0.25 (. ) ( ) [using clcultor] 3. 2 2x + 1 5 2 x log 2 2x + 1 log 5 2 x (2x + 1) log 2 (2 x) log 5 (2x + 1) 0.301 (2 x) 0.699 [Tke logs of both sides] 0.602x + 0.301 1.398 0.699x [removing brckets] See Growth nd decy 1.301x 1.097 x 0.843 Exmple The number of bcteri present in smple is given by N 800e 0.2t, where t is in seconds by Find () the initil number of bcteri (b) the time when the number of bcteri reches 000 () The initil number of bcteri occurs when t 0 N 800e 0.2t 800e 0.2x 0 [Substitute t 0 in the eqution for N] 800e 0 800 The initil number of bcteri is 800. (b) The number of bcteri, N, is equl to 000. N 800e 0.2t 000 800e 0.2t [Substitute N 000 in the eqution for N] e 0.2t 12.5 e 0.2t ln(12.5) ln(e 0.2t ) [Tke the logrithm to bse e of both sides] ln(12.5) 0.2t t 12.6 It tkes 12.6 sec. For the number of bcteri to rech 000 See Exercise 2 14
Exercises Solve for x: 1. () 5 x 125 (b) 3 x 1 27 (c) 2 2x 1 128 (d) 3-4 (e) 9 2x + 1 27 x 2. () 5 x 12 (b) 2 x 3 9 (c) 7-2x 5 (d) 4 x + 2 Exercise 2 1. The decy rte for rdio-ctive element is given by R 400e -0.03t where R is the decy rte in counts/s t time t(s). Find: (i) the initil decy rte (ii) the time for the decy rte to reduce to hlf the initil decy rte. 2. The chrge Q units on plte of condenser t seconds fter it strts to dischrge is given by the formul Q Q 0 -kt If the initil chrge is 5076 units nd Q 1840 when t 0.5s find: (i) the vlue of k (ii) the time needed for the chrge to fll to 00 units. (iii) the chrge fter 2 sec. Answers 1 () x 3 (b) x 4 (c) x 4 (d) x - (e) x -2 2 () x 1.54 (b) x 6.17 (c) x -0.41 (d) x - 1.67 Exercise 2. 1 (i) Ro 400 (ii) t 23.1s 2 (i) k 0.881 (ii) t 0.8s (iii) Q 87.8 units 15
SIMPLIFYING SURDS Definition A surd is n irrtionl number resulting from rdicl expression tht cnnot be evluted directly. For exmple 2, 3,, 20, 2,, 8, etc re ll surds, but 1,,, 8, 16, 64 re not. Surds cnnot be expressed exctly in the form eg: 2 1.414 See, nd cn only be pproximted by deciml. Simplifying Surds If surd hs squre fctor ie 4,, 16, 2, 36, then it cn be simplified. Exmples 1. 200 0 2 0 2 [NB: It is best to find the lrgest squre number fctor!] 2 2. 348 316 3 316 3 3 3 4 3 123 3. See Exercise 2 Addition nd Subtrction Only like surds cn be dded or subtrcted Exmples 1. 8-3 5 2 3 + 5-3 5-83 Sometimes it is necessry to simplify ech surd first 18 8 20 2 4 2 4 32-22 2 2 2 16
NB: 12 becuse re not like surds. See Exercise 3 Multipliction of Surds To multiply surds the whole numbers re multiplied together, s re the numbers enclosed by the rdicl sign Exmples: 2. 33 4 121 Sometimes it is possible to simplify fter multipliction 3. 2 6 1460 144 1 14 21 281 See Exercise 4 Division of Surds Expressions such s my be expressed s frctions nd simplified using Exmples 1. 2. 3 See Exercise 5 8 22 17
Exercises Decide whether the following rdicl expressions re rtionl or irrtionl nd evlute exctly if rtionl: () 2 (b) 2.2 (c) (d) 64 (e) 12 18 Exercise 2 Simplify () 20 (b) 48 (c) 00 (d) 212 (e) Exercise 3 Simplify () 23 3 (b) 3 3 (c) 2 32 4 (d) 4 24 Exercise 4 Simplify () 3 (b) 2 3 (c) 3 (d) 28 2 0 2 Exercise 5 Simplify () (c) (b) (d) Answers () 5 rtionl (b) 1.5 rtionl (c) irrtionl (d) 4 rtionl (e) irrtionl Exercise 2 () 2 (b) 43 (c) (d) 43 (e) Exercise 3 () 73 (b) (c) 42 3 (d)6 Exercise 4 () 30 (b) 21 (c) 30 (d) 802 Exercise 5 () 2 (b) 4 (c) 6 (d) 2 18
OPERATIONS ON SURDS Expnsion of Brckets The usul lgebric rules for expnsion of brckets pply to brckets contining surds (b + c) b + c nd ( + b)(c + d) c + bc +d + bd Exmples 1. 2(2 ) 2 2 2. 23(3 32) 2 66 6-66 3. ( ) ( )( ) 4 54-14 4. (6 43)(22 3 ) 212 86 330 121 24 3 86 330 121 2 2 3 86 330 121 43 86 330 121 See Rtionlizing surds Sometimes frctions contining surds re required to be expressed with rtionl denomintor. Exmples 1. 2. Conjugte surds The pir of expressions b nd b re clled conjugte surds. Ech is the conjugte of the other. The product of two conjugte surds does NOT contin ny surd term! ( b)( b) b Eg: ( 3)( 3) () (3) 3 7 We mke use of this property of conjugtes to rtionlize denomintors of the form b b nd 19
Exmple ( ) See Exercise 2 Opertions on frctions tht contin surds When dding nd subtrcting frctions contining surds it is generlly dvisble to first rtionlize ech frction: Exmple ( ) ( ) See Exercise 3 Exercises Expnd the brckets nd simplify if possible () 2(2 8) (b) (2 3)(3 4) (c) (1 ) (d) (11 3)(11 3) Exercise 2 Express the following frctions with rtionl denomintor in simplest form: 1. () (b) (c) 2. () (b) (c) Exercise 3 Evlute nd express with rtionl denomintor Answers () 2 82 (b) -5-23 (c) 11+2 (d) 2 Exercise 2 1. () (b) (c) 3 2. () Exercise 3 43 + 7 (b) 3 2 (c) -7-43 20
SCIENTIFIC NOTATION Scientific Nottion (or Stndrd Form) is wy of writing numbers in compct form. A number written in Scientific Nottion is expressed s number from 1 to less thn, multiplied by power of eg 2.3-3 To write number in Scientific Nottion: move the deciml point so tht there is one digit (which cnnot be zero), before the deciml point. multiply by power of, equl to the number of plces the deciml point hs been moved. If the deciml point is moved to the left, the power of is positive If the deciml point is moved to the right the power of is negtive Exmples 1. Write 5630 in Scientific Nottion. 5630 5.63 00 5.63 3 [remember:00 3 ] Move the deciml point three plces to the left. The number becomes 5.63 nd the power of is +3. 2. Write 0.00725 in Scientific Nottion. 0.00725 7.25 0.001 7.25-3 [remember 0.001-3 ] Move the deciml point three plces to the right. The number becomes 7.25 nd the power of is 3. 3. Clculte 3.5 5 5 18 3.5 5 5 18 3.5 5 5 + 18 [Clcultions cn be simplified by using index lws] 17.5 23 1.75 24 Alwys check your nswer: If the mgnitude of the number is less thn one, then the power of will be negtive. If the mgnitude of the number is greter thn or equl to then the power of will be positive. The mss of the moon is: 73 600 000 000 000 000 000 000 kg. 7.36 22 kg. The chrge on one electron is: 0.000 000 000 000 000 000 160 2 C 1.602-19 C Note how the size of number is more esily seen when written in scientific nottion. Errors re less likely when writing very big nd very smll numbers if they re in scientific nottion. See Exercises 21
Exercises Write the following numbers in Scientific Nottion. () 58000 (b) 0.0026 (c) 70.6 (d) 0.3 (e) 2 400 000 (f) 0.000 000 684 (g) 0.0704 (h) 0.260 (i) 17600 (j) 0.04080 (k) 0.000500 (l) 357 000 000 000 Exercise 2 Write the following numbers in Scientific Nottion () 6 370 000 m (the men rdius of the Erth) (b) 86 400 s (the time for the Erth to orbit the sun) (c) 0.0018 s (the hlf-life of rdioctive isotope) (d) 380 000 000 m (the verge distnce of the moon form the Erth) (e) 0.000 000 000 001 1 s (the time for ry of light to pss through window pne) (f) 637 000 000 000 000 000 000 000 kg (the mss of the plnet Mrs) Exercise 3 Write the following numbers in deciml form or in whole numbers. () 7 2 (b) 4 0 (c) 3.46 3 (d) 5.96-5 (e) 9 7 (f) 3.98 1 (g) 2.78 5 (h) 6.78-1 Answers () 5.8 4 (b) 2.6-3 (c) 7.06 (d) 3-1 (e) 2.4 6 (f) 6.84-7 (g) 7.04-2 (h) 2.60-1 (i) 1.76 4 (j) 4.08-2 (k) 5.00-4 (l) 3.57 11 Exercise 2 () 6.37 6 m (b) 8.64 4 s (c) 1.8-3 m (d) 3.8 8 m (e) 1.1-12 s (f) 6.37 23 kg Exercise 3 () 700 (b) 4 (c) 3460 (d) 0.0000596 (e) 90000000 (f) 39.8 (g) 278000 (h) 0.678 22
SIGNIFICANT FIGURES In scientific mesurements, significnt figures re used to indicte the ccurcy of mesurement. The lst digit in mesurement is often n estimte, good guess. All the digits in mesured vlue, including the lst estimted digit, re clled significnt figures or significnt digits. The number of Significnt Figures in mesured vlue Rule 1: Any non-zero digit is significnt. The position of deciml point mkes no difference. Exmple 1 15.7 3 sig. figs 157 3 sig. figs 1.57 3 sig. figs 2.7942 5 sig. figs Rule 2: Zeros between numbers re significnt. Exmple 2 1.05 3 sig. figs.51 4 sig. figs 200.708 6 sig. figs Rule 3: Zeros t the right hnd end of whole numbers re not significnt, unless otherwise stted. Exmple 3 70 1 sig. figs 2860 3 sig. figs 15090 4 sig. figs Rule 4: Zeros t the left hnd end of deciml numbers re not significnt. Exmple 4 0.28 2 sig. figs 0.0039 2 sig. figs 0.0604 3 sig. figs Rule 5: Zeros t the right hnd end of deciml numbers re significnt. Exmple 5 12. 0 3 sig. figs 0.760 3 sig. figs 0.48300 5 sig. figs 2.07090 6 sig. figs Significnt Figures nd Scientific Nottion The problem of deciding how mny significnt figures vlue hs is simplified by writing its vlue in Scientific Nottion. Exmine the number of digits in the first number below, not in the power of ten. () 0.0003 3-4 1 sig figrule 4 (b) 720000 7.2 5 2 sig figs Rule 3 (c) 660 6.6 2 2 sig figs Rule 3 (d) 660.0 6.600 2 4 sig figs Rule 5 (e) 0.66000 6.6000-1 5 sig figs Rule 5 (f) 808.01 8.0801 2 5 sig figs Rule 2 See 23
Rounding Off Sometimes, numbers must be rounded off so tht vlues do not pper to hve more significnt figures thn would be pproprite or relistic. Rule 1: Rule 2: If the lst digit is greter thn 5, the second lst digit increses by 1 when rounding off. Exmple 1 5.78 rounded off to 2 sig figs becomes 5.8 5.8 rounded off to 1 sig fig becomes 6 (not 6.0) If the lst digit is less thn 5, the second lst digit does not chnge. Exmple 2 1.304 rounded off to 3 sig figs becomes 1.30 1.30 rounded off to 2 sig figs becomes 1.3 1.3 rounded off to 1 sig fig becomes 1. Rule 3: If the lst digit is 5, the number is rounded off to the nerest even number. Exmple 3 4.85 rounded off to 2 sig figs becomes 4.8 4.55 rounded off, becomes 4.6 See Exercise 2 Write ech of the following mesurements in Scientific Nottion nd stte the number of significnt digits in the vlue. () 345 (b) 0.000306 (c) 870 (d) 20000 (e) 140.600 (f) 7.0 (g) 0.04080 (h) 0.0050 Exercise 2 Round off ech of the following numbers to one less significnt figure nd then to two less significnt figures. Exmple: 6.549 6.55 6.6 () 7.668 (b) 0.0854 (c) 21.092 (d) 255.6 (e) 6.75 (f) 2.85 Answers () 345 3.45 2 3 sig figs (b) 0.000306 3.06-4 3 sig figs (c) 870 8.7 2 2 sig figs (d) 20000 2 4 1 sig fig (e) 140.600 1.40600 2 6 sig figs (f) 7.0 7.0 2 4 sig figs (g) 0.04080 4.080-2 4 sig figs (h) 0.0050 5.0-3 2 sig figs () 7.668 7.67 7.7 (b) 0.0854 0.085 0.08 (c) 21.092 21.09 21.1 (d) 255.6 256 260 (e) 6.75 6.8 7 (f) 2.85 2.8 3 24