# 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 never-ending, non-repeting 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 y-is 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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