Logrithms LOGARITHMS www.mthletis.om.u
Logrithms LOGARITHMS Logrithms re nother method to lulte nd work with eponents. Answer these questions, efore working through this unit. I used to think: In the epression 6, wht re the terms for the, the nd the 6? Whih of the following hs the sme vlue s 9+ : or 6? If is positive, wht vlues of use to e negtive? Answer these questions fter ou hve worked through this unit. But now I think: In the epression 6, wht re the terms for the, the nd the 6? Whih of the following hs the sme vlue s 9+ : or 6? If is positive, wht vlues of use to e negtive? Wht do I know now tht I didn t know efore? 00% Logrithms Mthletis 00% P Lerning K
Logrithms Bsis Wht re Logrithms? A rithm is n eponent (or inde) in n epression. Let's s Inde Bse Bse numerl Insted of writing 'The required eponent of to find is ', it is esier to write: Bse numerl Inde Bse The mthemetil phrsing is: "Log se of equls ". Here re some emples: Rewrite these epressions with rithms 8 6 ` 8 ` 6 6 d 8 ` 6 ` 8 e - f - 9 ` ` j- ` - ` j 9 g ` ` h 7 ` 7 ` 7 i Solvefor if 6 ` 6 ` 6 j Solvefor if 6 ` 6 8 ` 8 K 00% Logrithms Mthletis 00% P Lerning
Logrithms Questions Bsis. Write these s rithms: 6 6 7 9 d e 0 000 f - g 8 9 h i 7 j 6. Solve for : 8 d 8 e f 00% Logrithms Mthletis 00% P Lerning K
Logrithms Questions Bsis. Evlute these rithms to single numer. 9 0000 6 d 66 e 97 f g ` j h ` 6 j. If 0 0 then: Find 0 Aording to the zero inde lw 0 for n. So find in the following: 0 0 K 00% Logrithms Mthletis 00% P Lerning
Logrithms Knowing More Logrithms re lled 's' for short. Simplifing s is not lws es to see, so some rules re needed to help. Log Lws Log Lws re rules to help simplif rithms. These rules re: The Multiplition Rule: + The Division Rule: - The Eponent Rule: n n The Chnge of Bse rule: m m The Multiplition nd Division Rule Multiplition Rule Division Rule + - Let p nd q p ` nd ` # ` ` p+ q + p p+ q q q Proof Let p nd q p ` nd p q ` ' p- q ` ` p-q - q Proof Here re some emples. Simplif the following 0 + 0 0 0 7-6 6 0 ^0 # 0h Multiplition rule 7 6 ` j Division rule 0 000 66 00% Logrithms Mthletis 00% P Lerning K
Logrithms Knowing More The Eponent Rule n n n ( # # #... # ) Proof n times + +... + n times n Simplif the following 8 0+ 0 8 0 + 0 6 0 + 0 0 ^ # h 0 00 Here re some other rules whih re found from eponents: Sine it's es to see the lw So the of numer the sme s it's se is. Sine 0 for n, it's es to see tht for n 0 So the (n se) of is 0. Simplif the following Sme numer 0 + d + -66 + ^h + 0- + 0- - 6 K 00% Logrithms Mthletis 00% P Lerning
Logrithms Knowing More Roots in Logs The rule for eponents n n is used when roots re in s. Here re some emples: Simplif these s # n n 6 6 6 6 6 66 # n n Using Lws to Epnd nd Simplif Epressions Log lws re used to epnd or simplif epressions. Here re some emples: Simplif these to one rithm with no surds + + d 6 + 6-6 w + ^h + d 6 6 6 + - w + + d 6 6 + - 6w # ( # # d ) 6 6 + - 6w ( d) ^w h 6 Epnd these rithms without eponents w m ( w) - 0 ` + 0 00 j 0 0 ` + 00 j w + - 0^0+ h - 000 w+ - 0^0+ h - 00 w- + 0^0+ h - 00% Logrithms Mthletis 00% P Lerning K 7
Logrithms Knowing More Clulting Using Log Lws Log Lws need to e omined sometimes to lulte s. Simplif these epressions 68-9 ^# h - 8-99 86-8 - 8 8 9 ^# h 88- ` j+ ` j + ` j + ` # j - 0-6+ 0-0000 d 6-8 ` j + 0-0 6 9+ 0-0 + 0 ` j 0 6-8 6 8 ` # - # j ` j - + -^# h ^ h+ ^ h ^ h- ^ h 6-8 K 00% Logrithms Mthletis 00% P Lerning
Logrithms Questions Knowing More. Use the Multiplition nd Division Rules to Simplif these rithms: 6+ 67+ 6 67+ 6-6 - + 8- d 8-0+ +. Use the Eponent Rules to simplif these: 8 0+ 00 8+ 8 d - 00% Logrithms Mthletis 00% P Lerning K 9
Logrithms Questions Knowing More. Use the lw for roots in s to simplif the following: 7 + 8 9 8. Simplif these to one rithm: 06+ 0-0 + - w - d + - ^+ h e - f - + 0 K 00% Logrithms Mthletis 00% P Lerning
Logrithms Questions Knowing More. Epnd these rithms, without eponents: w 0 ` j d + 6. Epnd these rithms s muh s possile. ` 9 j - 8 00% Logrithms Mthletis 00% P Lerning K
Logrithms Using Our Knowledge Logs with Frtions Rememer the eponent lw: Here re some emples: Write these without frtions n -n 6 - - 00 0 d ` j - # 0 - This rule is used to simplif s with frtions. Here re some emples: Simplif these rithms 8 8 - - - - 0 000 0 0 0^# 0 - h d 8-0 8 0 0+ 00 - - 0- - K 00% Logrithms Mthletis 00% P Lerning
Logrithms Using Our Knowledge Chnge of Bse There is lw for s whih llows us to hnge the se: m m Old se is New se is m Let Proof ` ` m m Find m of oth sides Eponent Rule ` m m ` ` m m m m The hnge of se lw n e used to find nother rule: m m / m m ` Here re some emples how to use this lw. Simplif these rithms 8 8 6 6 66 6 6 66 d 86 8 6 # 8 8 6 8 # 8 # 00% Logrithms Mthletis 00% P Lerning K
Logrithms Questions Using Our Knowledge. Simplif these rithms: 6 d e f. Use hnge of se to simplif these rithms: 7 9 6 6 d K 00% Logrithms Mthletis 00% P Lerning
Logrithms Questions Using Our Knowledge. Simplif these rithms: 7 k 9 ' k7 m # m d e 6# 6 f # 00% Logrithms Mthletis 00% P Lerning K
Logrithms Thinking More Eponentil Equtions Eponentil equtions re equtions with vrile in the eponent. These re solved finding ommon se on oth sides nd mking the eponents equl. Solve for the vrile in the following 8 + 6 p+ p- + Common se ` ` + ` Common se Common se p + p - ^ h ^ h p+ 0p-0 ` ` p+ 0p- 0 ` p Eponentil Equtions with Unommon Bses Sometimes ommon ses n't e found. If so, the utton on the lultor is used. On the lultor, the "" utton is " 0 " ( se 0) Solve for the vrile in the following 9-0 ` 9 ` - 0 09 Use the utton 00 ` ` Use the utton - 0 on the lultor 0 on the lultor 09. f ` -. f 00. f 0. 698f 7. ^d.p. h `. ^d.p. h m+ -m 7 m ` 7 77 + -m Find 7 ` ^m+ h7 - m 0 ` ^m+ h m - m 7 ` m 0 0 m+ m - 7 m 7 0 0 ` m + m - m Common se 07 07 ` m^067.... + h 966. 0 0 0 Eponent rule Use "" utton ` m. 66077. 6. (tod.p.) 6 K 00% Logrithms Mthletis 00% P Lerning
Logrithms Questions Thinking More. Solve these eponentil equtions: 6 6 7 d + e + 9 7 f + - 9 00% Logrithms Mthletis 00% P Lerning K 7
Logrithms Thinking More Logrithmi Equtions Logrithmi equtions hve the vrile inside the rithm. Log lws re used to get the vrile itself. Solve for the vrile in these equtions 6 + m 8 ` + ` m 8 ` ` + ` m ` 0 0 ` ` m ` 0 0. 698f ` 0 008. f 0 008. f. ^d.p. h Appling Logrithms Logrithms re in rel life formuls, nd so the n e used to solve rel prolems. Finding sound level Sound level is mesured in deiels (db) nd lulted using the formul db 60+ 0 0P where P is the sound intensit. - Find the sound level of sound with sound instensit P # 0 db 60+ 00^# 0 60+ 00+ 0 00 60 + ^. 77fh - 0.77 ^ d.p. h - h - Find the sound intensit (P) if the the sound level is 0dB. ` 0 60 + 0 0 ` P P - 0-0 P 8 K 00% Logrithms Mthletis 00% P Lerning
Logrithms Questions Thinking More. Use our lultor to find the vriles in these equtions orret to deiml ples. 0 6 m + 0 d k- 7 k e p- p f - + 9 00% Logrithms Mthletis 00% P Lerning K 9
Logrithms Questions Thinking More. The level of idit of liquid is mesured using the ph-sle. To find the level this formul is used: where H + is the onentrtion of the hdrogen ions. ph - 0 6 + H @ + - Find the ph-level of liquid with H 0 6 units. Find the ph-level of liquid with H + 7. # 0 - units. A sustne is neutrl if ph 7. Find H + of neutrl sustne. 0 K 00% Logrithms Mthletis 00% P Lerning
Logrithms Thinking More Eponentil Grphs Eponentil grphs re of funtions of the form or ` where j. The hve this form: ` - j ^,h ^-,h Here re some importnt properties out eponentil grphs: The lws ut the -is t ^0,h sine o for n vlue of. The eponentil grph never uts the -is sine is never negtive or zero if 0. The greter the vlue of (the se), the steeper the urve. Sketh the grphs of nd ` j on the sme set of es ` j ^-,h ^,h The -interept of ALL eponentil urves is lws ^0,h ^-, 0. h ^,0.h - - 0 No -interepts 00% Logrithms Mthletis 00% P Lerning K
Logrithms Thinking More The grphs elow re of the funtions nd 9 (Steeper urve) 8 7 6 (Gentler urve) - - - - - 0 - Whih is the steeper urve? is steeper thn. This is euse. ( will grow quiker thn ) Wht is the -interept of eh urve? Both urves hve -interept ^0,h Wh do oth urves hve the sme -interept? An eponentil urve will hve -interept sine 0. d Do either of the urves ever touh the -is? No, the urves get ver lose to the -is ut never touh. This is euse there is no vlue for suh tht or is negtive or zero. K 00% Logrithms Mthletis 00% P Lerning
Logrithms Questions Thinking More. The urve elow represents. Find the missing vlues in the sketh: (, d ) d (, ) (, ) e f (-, e ) (0, ) (-, f ). Without skething the grphs, identif the -interepts of 6 nd 0. How do ou know this? 6. The two urves elow represent nd 8. Identif eh grph nd nswer these questions: Identif the oordintes of eh point D(, ) A B B (, ) E(, ) C D A C(, ) E Wh is A ommon on oth urves? 00% Logrithms Mthletis 00% P Lerning K
Logrithms Questions Thinking More 7. The grph elow represents ` j. C( -, ) Identif the oordintes of eh point. A B B( -, ) A D(, ) C D Wht re the interepts of the eqution ` j? K 00% Logrithms Mthletis 00% P Lerning
Logrithms Thinking More Logrithmi Grphs The rithmi funtion is the inverse (opposite) funtion of. This mens the grph of is refletion of the eponentil grph out the line. If ^,h ^,h The properties of the rithm urve re the inverse of the eponentil urve: The lws uts the -is t ^,0h sine 0 for n vlue of. The rithmi grph never uts the -is sine the -vlue n not e negtive or zero if 0. The greter the vlue of (the se), the gentler the urve. Drw the urve of The urve of is the inverse of. Thus is the refletion of out the line. 8 ^,8h 7 6 ^-, 0. h 0 - - ^, h ^,h 6 7 8 ^0.,-h ^,h ^,h ^8,h Sine is the inverse of, if goes through n point (,) then will go through (,). For emple, the eponentil psses through (,) so the rithm psses through (,). 00% Logrithms Mthletis 00% P Lerning K
Logrithms Thinking More Here is n emple of two grphs: The grphs elow re of nd 6 ^8,h ^,h ^9,h - 0 ^,0h 0 ^,h ^,h 6 7 8 9 - - - Whih urve is steeper? The urve of is GENTLER thn the urve of. Rememer, with urves, the greter the vlue of, the gentler the urve. Wht is the -interept of eh of the urves? Both urves hve -interept (,0). All urves hve this interept. Do either of the urves ever touh the -is? No, the urves get ver lose to the -is ut never touh. This is euse n't e negtive or zero in or. 6 K 00% Logrithms Mthletis 00% P Lerning
Logrithms Questions Thinking More 8. The grph elow represents nd the dshed line represents. Find the oordintes of A, B, C nd D. A ^,h B -, ` j A B D C C D Whih funtion hs -interept nd whih hs n -interept? Wht re the interepts? 9. One of the lines elow represents nd the other represents 7: Wht hppens to the steepness of urve s inreses? Identif whih urve represents nd whih represents 7. Wht re the interepts of eh grph? 00% Logrithms Mthletis 00% P Lerning K 7
Logrithms Answers Bsis: Knowing More:. 6 6. 6 7 9 d e g i 0 000 f ` j- 9 8 h 7 j 6. e 6 0 d w 6 f m ^+ h. e 7 d 9 f. w + + 0 + 0 + 0. 9 0 000 - e g 6 d 6 6 7 9 f h 6-6. d - ^+ h + -. 0 0 ^- h+ ^+ h- ` 0 0 `. Using Our Knowledge: - - 0 ` 0 ` 0 e - - d f - -. Knowing More: d. d. d 8 K 00% Logrithms Mthletis 00% P Lerning
Logrithms Answers Using Our Knowledge: Thinking More:. d 6. A is ommon on oth urves euse the -interept on n eponentil grph is lws (0,) e f 7. A ^0, h C ^-9, h B ^-, h D, ` j. Thinking More: -interept is (0,). There re no -interepts. d e f 8. A ^0, h B ^0, h. 6. 08. C `,- j D ^, h e m 066. d k -9. p. ( d.p.) f.9 ( d.p.) hs -interept t (0,) hs n -interept t (,0). ph 6 9. As inreses, the urve eomes gentler, the steepness of the urve lessens. ph.77 ( d.p.) The non-dotted urve represents + - H 0 7 (,0) for oth grphs. e f d. Both (0,) s nthing to the power of 0 is. 6. A ^0, h B ^8, h C ^, h D ^6, h E ^6, h 00% Logrithms Mthletis 00% P Lerning K 9
Logrithms Notes 0 K 00% Logrithms Mthletis 00% P Lerning
Logrithms Notes 00% Logrithms Mthletis 00% P Lerning K
Logrithms Notes K 00% Logrithms Mthletis 00% P Lerning