Week 8. Topic 2 Properties of Logarithms


 Abner Cooper
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1 Week 8 Topic 2 Popeties of Logithms 1 Week 8 Topic 2 Popeties of Logithms Intoduction Since the esult of ithm is n eponent, we hve mny popeties of ithms tht e elted to the popeties of eponents. They e listed hee, long with few emples nd cutions, ut the wy to ecome mste of ithmic popeties is to pctice using those popeties. The ook hs n epnsive polem set, so do s mny s you cn. Net week, we ll use these popeties to solve equtions, dding yet nothe level of compleity. The tetook (pges ) is petty good hee, with sevel good emples. Reding Fist, hee gin is the definition of ithm: Logithm Definition ( M ) = y is equivlent to y M = Logithms hve sevel useful popeties, ll the esult of this definition. A couple of the popeties e stightfowd: (1) = 0 mkes sense ecuse 0 = 1 () = 1 mkes sense ecuse 1 = Rememe to ing the se ove nd ump up the othe side of the eqution while the potion dops out.
2 Week 8 Topic 2 Popeties of Logithms 2 The 5 popeties of ithms tht equie moe wok to pove e 1. M 2. ( ) = 3. M + N = ( M N ) 4. ( M ) ( N) = 5. ( M ) = ( M ) Thee is summy pge of these t the end. Hee, let s look t ech of these in tun. Popety 1: M In Topic 1, we found this ws tue y popeties of invese functions, ut we didn t pove it. Let s use simple tick to pove it hee. Fom the definition, use ( M ) = y, nd then sustitute y into Popety 1: y Now let s ewite this with ithm of se : y = M Tht line sys ectly wht we just sid few lines go, so this popety is tue. 5 (2 7) Emple: Popety 2: ( ) = We sw this in Topic 1 s well, ut with diffeent lettes. If you eponentite nume, then tke the ithm of it (with the sme se), you get ck the oiginl nume. This is tue y the definition of the ithm. Just move the se of the ithm to ump up the othe side of the eqution () into the eponent, nd you ve shown Popety 2. Emple: ln( e )
3 Week 8 Topic 2 Popeties of Logithms 3 Popety 3: M + N = ( M N ) This popety comes fom the popeties of eponents. Let s look t n emple fist to veify this is tue. Suppose M = 1000, N = 10,000, nd = 10. We sustitute to find: (1000) + (10, 000) = (10, 000, 000) = 7 We got to 10,000,000 y multiplying the numes, ut we could lso hve done , nd then we dd the eponents to get When we multiply powes of the sme se, we dd the eponents. Since the ithm picks the eponent off the tees, we could lso do the ddition fte getting the eponents. A poof is in the ook, if you e cuious. It s fom the definition of the ithm gin. Emples: (2) + ( ) (2 ) Used to comine ithms (10 ) 1+ ( ) Used to tun poduct into sum Cution! Multipliction inside ecomes ddition outside. You cn t simplify ( + 10). Popety 4: ( M ) ( N) = When you divide powes of the sme se, you sutct the eponents. So if we wote oth M nd N s powes of, we could pove this popety quickly. Tht s in the ook s well. Emples: 2(50) 2(5) 2(10) ln(20 ) ln(100) = ln 5 Cution! Division inside ecomes sutction outside. A LOT of people wnt to chnge this popety to something like ( M ), ut tht is not coect. The division is inside. ( N )
4 Week 8 Topic 2 Popeties of Logithms 4 Popety 5: ( M ) = ( M ) This is lso elted to popeties of eponents. Specificlly, when you ise powe to powe, you multiply the eponents. The ook hs ief poof of this s well. Emples: (40 ) (40) 2 ( ) 2( ) Hee is n emple comining popeties. Notice the eponent only pplies to the 2, not to the 3. So we cn t use Popety 5 until we use Popety 3: (3 2 ) = (3) + (2 ) = (3) + (2) O we cn go the othe wy to put ithms togethe, s long s we e ceful out wht the eponents pply to: 2ln( y) 4ln(3 ) = ( ) 2 ln( y ) ln 3 2 y = ln 4 ( 3) Notice the fouth powe pplied to eveything inside the second ntul ithm: oth the 3 nd the. 4 Chnge of Bse Fomul If ithm hs se you don t like, you don t hve to put up with it! ( ) ( ) = ( ) Usully you use this to wite n epession with se 10 o se e. Fo emple: ( ) ln( ) 5( ) = o 5( ) = (5) ln(5)
5 Week 8 Topic 2 Popeties of Logithms 5 OnetoOne Popety When we tke sque oot to solve squed eqution, we hve to include ± on the solution. Tht s ecuse the sque function is not onetoone. The sque of 2 is the sme s the sque of 2. Howeve, oth ithmic nd eponentil functions e onetoone. Tht leds to tidy esult: If ( ) = ( y), then = y If y =, then = y The fist line sys tht you cn t two diffeent numes nd get the sme nswe. The second line is simil sttement, ut with eponentition. Conclusions In this couse, these popeties e mostly used to solve equtions, which we ll do net week. But even with tht, ithms my not seem vey useful. The ook hs few ppliction polems, ut the elity is tht eponentil gowth is eveywhee: ny time quntity inceses y fied pecentge ech time peiod. To solve equtions o lgeiclly mnipulte epessions tht hve viles in eponents, we need ithms. A pcticl use of ithms tht I implemented fo jo ws to hndle etemely lge numes 1. If you wnt to compute y = these dys, Ecel o clculto will do tht fo you esily. But ck then, it ws cusing polems ecuse the esult is so close to zeo. My net step ws to multiply this y vey lge nume so the finl esult would e etween 0 nd 1, ut the softwe wsn t llowing it. I ewote the eqution s ln( y ) = 95 ln(0.03) + 90 ln(0.97). The compute did tht clcultion quite esily nd told me ln( y) , so y e Thee ws moe mth fte tht to get the finl nswe, ut I got ound the limittions of the compute using peclculus 2. The net two pges e efeence sheets suitle fo onepge pinting. The fist is list of the popeties fo use in this section. The second is list of the popeties tht we mostly use to solve equtions. Tht might come in hndy net week. 1 Stngely, I don t ememe wht I ws tying to ccomplish, only the mth I used. 2 Right fte this, I discoveed the softwe I ws using hd uiltin function to do these clcultions fo me.
6 Week 8 Topic 2 Popeties of Logithms 6 Usefulness of Logithm Popeties Popety Popety 1 M When would I use it? When you hve ithm in n eponent 5 6 Emple: 5 Popety 2 ( ) = When you e finding the ithm of n eponentil epession with the sme se Emple: 6 3(3 ) 6 Be ceful! To simplify (2 10 ), you hve to use Popety 3 fist! Popety 3 M + N = ( M N) When n epession o eqution hs two tems with ithms of the sme se Emple: (6) + (11) Be ceful! 2(8 + ) does not simplify! Popety 4 ( M ) ( N) = When n epession o eqution hs two tems with ithms of the sme se Emple: (200) (2) Be ceful! 2(8 ) does not simplify! Popety 5 ( M ) = ( M ) Chnge of Bse Fomul ( M ) = ( M ) ( ) To get n eponent out of the inside of ithm Emple: 2 (8 ) Be ceful! To simplify 2(3 8 ) you hve to use popety 3 fist! When n eqution o epession hs ithms with two diffeent ses Emple: 5( ) = 2( ) 1
7 Week 8 Topic 2 Popeties of Logithms 7 Popeties of Logithms Useful fo Solving Equtions SumofLogs Rule ( MN) = ( M ) + ( N) DiffeenceofLogs Rule = ( M ) ( N) Emple: (1, 000, 000) = (100) + (10, 000) 6 = Emple: 64 2 = 2(64) 2(16) 16 2(4) = 6 4 Powe Rule ( M ) = ( M ) ChngeofBse Fomul ( M ) ( M ) = ( ) Emple: (100 ) = (100) = 2 Emple: (100) 3(100) = (3) Definition of Logithm = y () = y Emple: 3 () = is equivlent to = OnetoOne Popeties If = y, then = y If () = (y), then = y Emples: 3 = 3 2 = 2 (100) = ( 2 ) 100 = 2
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