Secion. Logarihmic Properies 59 Secion. Logarihmic Properies In he previous secion, we derived wo imporan properies of arihms, which allowed us o solve some asic eponenial and arihmic equaions. Properies of Logs Inverse Properies: Eponenial Propery: r A r A Change of Base: c ( A) A ( ) c While hese properies allow us o solve a large numer of prolems, hey are no sufficien o solve all prolems involving eponenial and arihmic equaions. Video Eample 1: Sum and Difference Properies of Logarihms. Properies of Logs Sum of Logs Propery: A C ( AC ) Difference of Logs Propery: A A C C I s jus as imporan o know wha properies arihms do no saisfy as o memorize he valid properies lised aove. In paricular, he arihm is no a linear funcion, which means ha i does no disriue: (A + B) (A) + (B). To help in his process we offer a proof o help solidify our new rules and show how hey follow from properies you ve already seen. Le a A and c C, so y definiion of he arihm, a A and c C
60 Chaper Using hese epressions, AC a c ac Using eponen rules on he righ, AC Taking he of oh sides, and uilizing he inverse propery of s, ac AC a c Replacing a and c wih heir definiion esalishes he resul AC A C The proof for he difference propery is very similar. Wih hese properies, we can rewrie epressions involving muliple s as a single, or reak an epression involving a single ino epressions involving muliple s. Eample 1 Wrie 5 8 as a single arihm. Using he sum of s propery on he firs wo erms, 5 8 5 8 0 This reduces our original epression o 0 Then using he difference of s propery, 0 0 0 Eample 5 wihou a calculaor y firs rewriing as a single arihm. Evaluae On he firs erm, we can use he eponen propery of s o wrie 5 5 5 Wih he epression reduced o a sum of wo s, 5 sum of s propery 5 ( 5) (100 ) Since 100 = 10, we can evaluae his wihou a calculaor: (100) 10, we can uilize he Try i Now 1. Wihou a calculaor evaluae y firs rewriing as a single arihm: 8
Secion. Logarihmic Properies 61 Eample (video eample here) y Rewrie ln as a sum or difference of s 7 Firs, noicing we have a quoien of wo epressions, we can uilize he difference propery of s o wrie y ln ln ln(7) 7 y Then seeing he produc in he firs erm, we use he sum propery ln y ln(7) ln ln( y) ln(7 ) Finally, we could use he eponen propery on he firs erm ln ln( y) ln(7) ln( ) ln( y) ln(7 ) Ineresingly, solving eponenial equaions was no he reason arihms were originally developed. Hisorically, up unil he adven of calculaors and compuers, he power of arihms was ha hese properies reduced muliplicaion, division, roos, or powers o e evaluaed using addiion, suracion, division and muliplicaion, respecively, which are much easier o compue wihou a calculaor. Large ooks were pulished lising he arihms of numers, such as in he ale o he righ. To find he produc of wo numers, he sum of propery was used. Suppose for eample we didn know he value of imes. Using he sum propery of s: ( ) ( ) () value (value) 1 0.0000000 0.01000 0.7711 0.600600 5 0.6989700 6 0.778151 7 0.850980 8 0.900900 9 0.955 10 1.0000000 Using he ale, ( ) ( ) () 0.01000 0.7711 0.778151 We can hen use he ale again in reverse, looking for 0.778151 as an oupu of he arihm. From ha we can deermine: ( ) 0.778151 (6). By doing addiion and he ale of s, we were ale o deermine 6. Likewise, o compue a cue roo like 8 1/ 1 1 ( 8) 8 (8) (0.900900) 0.01000 So 8. ( )
6 Chaper Alhough hese calculaions are simple and insignifican hey illusrae he same idea ha was used for hundreds of years as an efficien way o calculae he produc, quoien, roos, and powers of large and complicaed numers, eiher using ales of arihms or mechanical ools called slide rules. These properies sill have oher pracical applicaions for inerpreing changes in eponenial and arihmic relaionships. Eample Recall ha in chemisry, ph H. If he concenraion of hydrogen ions in a liquid is douled, wha is he affec on ph? Suppose C is he original concenraion of hydrogen ions, and P is he original ph of he liquid, so P C. If he concenraion is douled, he new concenraion is C. Then he ph of he new liquid is ph C Using he sum propery of s, ph C ( ) ( C) ( ) ( C) Since P C, he new ph is ph P ( ) P 0.01 When he concenraion of hydrogen ions is douled, he ph decreases y 0.01. Log properies in solving equaions The arihm properies ofen arise when solving prolems involving arihms. Eample 5 Solve ( 50 5) ( ). In order o rewrie in eponenial form, we need a single arihmic epression on he lef side of he equaion. Using he difference propery of s, we can rewrie he lef side: 50 5 Rewriing in eponenial form reduces his o an algeraic equaion: 50 5 10 100
Secion. Logarihmic Properies 6 Solving, 50 5 100 5 50 5 50 1 Checking his answer in he original equaion, we can verify here are no domain issues, and his answer is correc. Try i Now. Solve ( ) 1 ( ). More comple eponenial equaions can ofen e solved in more han one way. In he following eample, we will solve he same prolem in wo ways one using arihm properies, and he oher using eponenial properies. Eample 6a In 008, he populaion of Kenya was approimaely 8.8 million, and was growing y.6% each year, while he populaion of Sudan was approimaely 1. million and growing y.% each year. If hese rends coninue, when will he populaion of Kenya mach ha of Sudan? We sar y wriing an equaion for each populaion in erms of, he numer of years afer 008. Kenya( ) 8.8(1 0.06) Sudan( ) 1.(1 0.0) To find when he populaions will e equal, we can se he equaions equal 8.8(1.06) 1.(1.0) For our firs approach, we ake he of oh sides of he equaion 8.8(1.06) 1.(1.0) Uilizing he sum propery of s, we can rewrie each side, (8.8) 1.06 (1.) 1.0 Then uilizing he eponen propery, we can pull he variales ou of he eponen World Bank, World Developmen Indicaors, as repored on hp://www.google.com/pulicdaa, rerieved Augus, 010
6 Chaper (8.8) 1.06 (1.) 1.0 Moving all he erms involving o one side of he equaion and he res of he erms o he oher side, 1.06 1.0 (1.) (8.8) Facoring ou he on he lef, 1.06 1.0 (1.) (8.8) Dividing o solve for (1.) (8.8) 1.06 1.0 15.991years unil he populaions will e equal. Eample 6 (video eample here) Solve he prolem aove y rewriing efore aking he. Saring a he equaion 8.8(1.06) 1.(1.0) Divide o move he eponenial erms o one side of he equaion and he consans o he oher side 1.06 1. 1.0 8.8 Using eponen rules o group on he lef, 1.06 1. 1.0 8.8 Taking he of oh sides 1.06 1. 1.0 8.8 Uilizing he eponen propery on he lef, 1.06 1. 1.0 8.8 Dividing gives 1. 8.8 15.991 years 1.06 1.0
Secion. Logarihmic Properies 65 While he answer does no immediaely appear idenical o ha produced using he previous mehod, noe ha y using he difference propery of s, he answer could e rewrien: 1. 8.8 (1.) (8.8) 1.06 (1.06) (1.0) 1.0 While oh mehods work equally well, i ofen requires fewer seps o uilize algera efore aking s, raher han relying solely on properies. Try i Now. Tank A conains 10 liers of waer, and 5% of he waer evaporaes each week. Tank B conains 0 liers of waer, and 50% of he waer evaporaes each week. In how many weeks will he anks conain he same amoun of waer? Imporan Topics of his Secion Inverse Eponenial Change of ase Sum of s propery Difference of s propery Solving equaions using rules Try i Now Answers 1. 5. 1..187 weeks
66 Chaper Secion. Eercises Simplify o a single arihm, using arihm properies. 8 7 1.. 1. 7 1 10 5. 50 1. 5 (7) 6. 1 7 8 7. 1 5 6 8. 5 9. 10. ln ln 9 11. ln 6 ln 1. 1 1. 1 1. 1 15. y z 16. y z 1 Use arihm properies o epand each epression. 15 1 17. y 19 18. a 5 z c a 19. ln 5 c 0. ln a 5 c 1. y. y. ln y y 1 y. ln 1 5 5. y y 6. 7 y y 9
Secion. Logarihmic Properies 67 Solve each equaion for he variale. 7 96 7. 8. 7 5 7 9. 171.1 191.16 0. 01.07 81.1 1. 0.1 0.08 5e 10e. e e 0.09 0.1 7 6. ( ). 5. ln 1 6. ln 5 5 5 7. 8. 9. 0. 9 1. 1. 5. ( 1) 1. 6 6 ( ) ( ) 5 5. 1 1 6. 15 15 7. ln ln ln 7 8. ln ln 6 ln 6