Chapter 1: Logarithmic functions and indices


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1 Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) Hint: The mth root of. Use the rule m n m n to simplify the inde. r r Rewrite the epression with the numers together nd r r the r terms together. 6 r 6 6r 5 r r r c 4 4 Use the rule m n m n to simplify the inde Use the rule m n m n d e ( ) Use the rule ( m ) n mn to simplify the inde f ( ) 4 Use the rule ( m ) n mn to simplify the inde
2 Emple Simplify: 4 _ _ c ( _ ) d Use the rule m n m n. 7 Rememer ( ). _ _ This could lso e written s. _ _ Use the rule m n m n. c ( _ ) Use the rule ( m ) n mn. _ d Use the rule m n m n. _ _ _ Emple Evlute: 9 _ 6 4 _ c 4 9 _ d 5 _ 9 _ Using m m. 9 Both nd re squre roots of 9. 9 strictly mens nd 9 ut lwys check if the negtive squre root is required nswer. 6 4 _ 64 This mens the cue root of As c 49 _ ( 49 ) n Using m m n. 4 d 5 _ Using m 5 m. ( 5 ) 5 5 ( 5) 5 This mens the squre root of 49, cued.
3 Eercise A Simplify these epressions: 4 c 4p p d 4 e k k f (y ) 5 g 0 5 h (p ) p 4 i ( ) j 8p 4 4p k 4 5 l 7 4 m 9 ( ) n 4 6 o 7 4 ( 4 ) p (4y ) y q 6 5 r 4 5 Simplify: 5 7 c _ 5_ d ( ) _ e ( ) 5_ f g 9 _ 6 h 5 _ 5 _ 5 i 4 5 Evlute: _ 5 8 _ c 7 _ d 4 e 9 _ f ( 5) g ( _ 4 ) 0 h 9 6 _ 4 i ( j ( 7) _ 8 k ( 6_ ) 5 l ( 4 9 ) _ 6 5 ) _. You cn write numer ectly using surds, e.g., 5, 9. You cnnot evlute surds ectly ecuse they give neverending, nonrepeting deciml frctions, e.g The squre root of prime numer is surd. You cn mnipulte surds using these rules: _ () You cn rtionlise the denomintor of y multiplying the top nd ottom. y
4 Emple 4 Simplify: 0 c _ 94 c (4 ) 4 Use the rule Cncel y _ is common fctor. 6 ( ) Work out the squre roots 4 nd 49. 6(5 7) (8) 8 6 Emple 5 Rtionlise the denomintor of: Multiply the top nd ottom y. ( ) Multiply the top nd ottom y. Rememer 6 Simplify your nswer 4
5 Eercise B Simplify: _ _ You need to know how to write n epression s logrithm log n mens tht n, where is clled the se of the logrithm In the IGCSE the se of the logrithm will lwys e positive integer greter thn. Emple 6 Write s logrithm So log 5 Here, 5, n. Bse Logrithm In words, you would sy the logrithm of, to se, is 5. In words, you would sy to the power 5 equls. Emple 7 Rewrite s logrithm: c 0 04 log log c log
6 log 0 Becuse 0. log Becuse. Emple 8 Find the vlue of log 8 log c log ( 5 ) log 8 4 Becuse 4 8. log Becuse 4 _ c log ( 5 ) 5 Becuse 5 5! You cn use the log key on clcultor to clculte logrithms to se 0. Emple 9 Find the vlue of for which So log Since 0 00 nd 0 000, must e log somewhere etween nd..70 (to s.f.) The log (or lg) utton on your clcultor gives vlues of logs to se 0. Eercise C Rewrite s logrithm: _ 9 c d Rewrite using power: log 6 4 log 5 5 c log 9 _ d log 5 0. e log Find the vlue of: log 8 log 5 5 c log d log e log 79 f log 0 0 g log 4 (0.5) h log ( 0 ) 4 Find the vlue of for which: log 5 4 log 8 c log 7 d log () 6
7 5 Find from your clcultor the vlue to s.f. of: log 0 0 log 0 4 c log d log Find from your clcultor the vlue to s.f. of: log 0 log 0 5. c log 0 0. d log You need to know the lws of logrithms Suppose tht Rewriting with powers: log nd log y c nd c y Multiplying: y c c (see section.) y c Rewriting s logrithm: log y c log y log log y (the multipliction lw) It cn lso e shown tht: log ( y ) log log y (the division lw) log () k k log (the power lw) Rememer: c c c Rememer: ( ) k k Note: You need to lern nd rememer the ove three lws of logrithms. Since ( ), the power rule shows tht log ( ) log ( ) log. log ( ) log And from the previous section log (since ) log 0 (since 0 ) Emple 0 Write s single logrithm: log 6 log 7 log 5 log c log 5 log 5 d log 0 4 log 0 ( _ ) log (6 7) Use the multipliction lw. log 4 log (5 ) Use the division lw. log 5 c log 5 log 5 ( ) log 5 9 First pply the power lw to oth prts of log 5 log 5 ( ) log 5 8 the epression. log 5 9 log 5 8 log 5 7 Then use the multipliction lw. d 4 log 0 ( _ ) log 0 ( _ ) 4 log 0 ( log 0 log 0 ( 6 ) log 0 ( log ) Use the power first. ) 6 Then use the division lw. 7
8 Emple Write in terms of log, log y nd log z log ( yz ) log ( y ) c log ( z ) d log ( y 4 ) log ( yz ) c d log ( ) log y log (z ) log log y log z log ( y ) log log (y ) log log y log ( y z ) log ( y ) log z log log y log z log _ log y log z log ( 4 ) log log ( 4 ) log 4 log log 4 log. Use the power lw ( y y _ ). Eercise D Write s single logrithm: log 7 log log 6 log 4 c log 5 log 5 0 d log log 6 e log 0 5 log 0 6 log 0 ( _ 4 ) Write s single logrithm, then simplify your nswer: log 40 log 5 log 6 4 log 6 9 c log 4 log d log 8 5 log 8 0 log 8 5 e log 0 0 (log 0 5 log 0 8) Write in terms of log, log y nd log z: log ( y 4 z) log ( 5 d y ) c log ( ) log ( y z ) log 8
9 . 5 You cn use the chnge of se formule to solve equtions of the form Working in se, suppose tht: Writing this s power: Tking logs to different se : Using the power lw: log m m log ( m ) log m log log Writing m s log : This cn e written s: log log log log log log This is the chnge of se rule for logrithms. Using this rule, notice in prticulr tht log log log, ut log, so: log log Emple Solve the following equtions, giving your nswers to significnt figures. 0 8 c log 0 Use the definition of logrithms from section.. By chnge of se formul, chnging to se 0 log 0 log 0 0 log 0 log Some clcultors cn evlute log 0. If your clcultor does not hve this fcility, you cn use the chnge of se formul nd use se 0 The log utton on your clcultor uses log 0. Use this to find log 0 0 nd log 0. Give nswer to sf. 8 log 8 Use the definition from section.. Chnging to se 0 log 8 log 0 log 0 8 Evlute using clcultor nd give nswer to sf..5 c log This cn e found directly using the log utton on 0.55 clcultor. NB A logrithm cn give negtive nswer: log < 0 when 0 < < 9
10 Emple Solve the eqution log 5 6 log 5 5: 6 log 5 log 5 5 Use chnge of se rule (specil cse). Let log 5 y y 6 5 y y 6 5y Multiply y y. y 5y 6 0 (y )(y ) 0 So y or y log 5 or log 5 5 or 5 5 or 5 Eercise E Write s powers. Find, to deciml plces: log 7 0 log 45 c log 9 d log e log 6 4 Solve, giving your nswer to significnt figures: c 6 Solve, giving your nswer to significnt figures: 75 0 c 5 d Solve, giving your nswer to significnt figures: log 8 9 log log 4 log 4 0 c log log 4. 6 You need to e fmilir with the functions y nd y log nd to know the shpes of their grphs As n emple, look t tle of vlues for y : Hint: A function tht involves vrile power such s is clled n eponentil function. 0 y _ 8 _ 4 _ 4 8 Note tht 0 (in fct 0 lwys if 0) nd ( negtive inde implies the reciprocl of positive inde) 8 0
11 The grph of y looks like this: 0 Other grphs of the type y re of similr shpe, lwys pssing through (0, ). Now look t the tle of vlues of y log : _ 8 _ 4 _ 4 8 y 0 You should note tht the vlues for nd y hve swpped round. This mens tht the shpe of the curve is simply reflection in the line y =. y The grph of y log will hve similr shpe nd it will lwys pss through (, 0) since log 0 for every vlue of. y Hint: Notice tht log 0 O Hint: The y is is n symptote to the curve. Emple 4 On the sme es sketch the grphs of y y nd y.5 On nother set of es sketch the grphs of y _ ( ) nd y. For ll the three grphs, y when 0. 0 When > 0, > >.5 When < 0, < <.5 Work out the reltive positions of the three grphs y y y y.5 0
12 _ So y ( _ ) is the sme s y ( ). ( m ) n mn So the grph of y ( _ ) is reflection in the yis of the grph of y. y y ( ) y 0 Emple 5 On the sme es, sketch the grphs of y log nd y log 5. For oth grphs y = 0 when =. But log so y log psses through (, ) nd log 5 5 so y log 5 psses through (5, ). By considering the shpe of the grphs etween y = 0 nd y =, you cn see tht log > log 5 for >. Since the log grphs re reflections of the eponentil grphs then from Emple 4 you cn see tht the reverse will pply the other side of (, 0). So log < log 5 for <. Since log 0 for every vlue of y y log y log 5 Eercise F On the sme es sketch the grphs of y 4 y 6 c y ( _ 4 ) On the sme es sketch the grphs of y y log c y ( _ ) On the sme es sketch the grphs of y log 4 y log 6 4 On the sme es sketch the grphs of y y log c Write down the coordintes of the point of intersection of these two grphs.
13 Eercise G Simplify: y y 5 5 c (4 ) 5 d 4 4 Simplify: 9 (4 _ ) _ 4 d _ 6 _ Evlute: ( 8 _ 7 ) d ( 5 89 ) 4 Simplify: 6 5 Rtionlise: 5 5 _ Epress log (p q) in terms of log p nd log q. Given tht log (pq) 5 nd log (p q) 9, find the vlues of log p nd log q. 7 Solve the following equtions giving your nswers to significnt figures: Given tht log, determine the vlue of. Clculte the vlue of y for which log y log (y 4). c Clculte the vlues of z for which log z 4 log z. 9 Find the vlues of for which log log. 0 Solve the eqution log ( ) log 9 (6 9 ).
14 Chpter : Summry Logrithms You cn simplify epressions y using rules of indices (powers). m n m n m n m n m m m m n m m n ( m ) n mn 0 Chpter : Summry You cn mnipulte surds using the rules: The rule to rtionlise surds is: Frctions in the form, multiply the top nd ottom y. 4 log n mens tht n, where is clled the se of the logrithm. 5 log 0 log 6 log 0 is sometimes written s log. 7 The lws of logrithms re log y log log y log ( y ) log log y log () k k log 8 From the power lw, log ( ) log (the power lw) (the multipliction lw) (the division lw) 9 The chnge of se rule for logrithms cn e written s log log log 0 From the chnge of se rule, log log 4
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