Ojeives: n nlze nd inerpre he ehvior of rihmi funions, inluding end ehvior nd smpoes. n solve rihmi equions nlill nd grphill. n grph rihmi funions. n deermine he domin nd rnge of rihmi funions. n deermine he inverse funion of rihmi funion. n deermine regression models from d using pproprie ehno nd inerpre he resuls. n jusif nd inerpre soluions o ppliion prolems. Definiions / Voulr / Grphil nerpreion: Two ws o solve for vrile in he eponen re ) grphill using he lulor inerse funion; ) lgerill using rihms. A LOG S AN EXPONENT. The rihmi funion funion. is he of he eponenil Thus, he oupu of he rihmi funion is he eponen (inpu) of he eponenil funion. The se of he eponenil funion is he se of he rihmi funion. The inpu (rgumen) o rihmi funion nno e zero or negive. E : rewrie 3 9 in form: E : rewrie 5 in form: E 3: rewrie 3 3 in form: E : rewrie 5 5 in eponenil form: E 5: rewrie in eponenil form: E 6: Solving for vrile in he eponen using rihms 00 00(.05)
Rell he generl form of n eponenil funion: Where: = finl moun = iniil moun = growh for = ( + growh re) = ime Douling ime emple: Find he ime i kes $00 o doule n nnul ineres re of.5% Se up: 00 00(.05) nd solve for Hlf-life emple: Find he hlf-life of susne h is deing re of 0% per d. Se up: ( 0.) nd solve for NOTE: We do no need o know he iniil moun o find douling ime or hlf-life, we jus need o know he re he iniil moun is growing or deing. Bses nd Properies of Logrihms The wo mos ommon ses for rihms re he ommon se 0, nd he nurl se e. Properies of he Common Logrihm: Clulor LOG ke lules 0 mens 0 0 0 nd 0 The funions 0 nd re inverses: 0 ( ) for ll nd 0 for ll 0 (ll rgumens mus e posiive) For nd oh posiive nd ll : ( ) ( )
Properies of he Nurl Logrihm: ln mens e ln = 0 nd ln e = The funions e nd ln re inverses: ln( e ) for ll nd e ln for ll >0 For nd oh posiive nd ll : ln( ) ln ln ln ln( ) ln ln ln Chnge of Bse Formul To hnge n unommon se rihm o one whih is lulor friendl so h i m e evlued, we pill use ommon se for suh s se 0 or se e. Noe: he formul elow works for ll ses, so long s he ses on he righ side of he equion re he sme. Convering eween Periodi Growh Re nd Coninuous Growh Re An eponenil funion n e wrien s So we n eque or k k e ln ln e k k e E: Conver Q (. ) 5 ino he form Q k e Annul Growh Re: Coninuous Growh Re: Grphs of Eponenil nd Logrihmi Funions, Asmpoes, nd End Behvior Eponenil funions nd rihmi funions re inverses of eh oher. Thus, he oupu of he rihmi funion is he inpu o he eponenil funion; nd he inpu o he rihmi funion is he oupu of he eponenil funion. The grph of he rihmi funion is he grph of he eponenil funion refleed ou he line. The eponenil grph hs horizonl smpoe nd he rihmi grph hs veril smpoe. 3
The pren funion End ehvior of: f ( ) f ( ) ln hs veril smpoe 0 ln lim ln lim ln The pren funion End ehvior: lim e f ( ) f ) 0 ( e hs horizonl smpoe 0 e lim e Domin nd Rnge of Logrihmi Funions Noe: if rnsformion is pplied o he rihmi funion, he domin m hnge. For emple, if he funion is shifed horizonll, he limis ove will shif ordingl. The -vlue h mkes he rgumen of he 0 eomes he veril smpoe. The rihmi funion does no hve horizonl smpoe. E: The veril smpoe of he funion f ( ) (3 ) is Generl ehvior: f ( ) ln, 0 f 0 hen he funion is inresing (slowl) nd onve down. f 0 hen he funion is deresing (slowl) nd onve up.
Logrihmi Models nd Sles ph Sle (Chemil Aidi) ph [ H ] where H is he hdrogen ion onenrion given in moles per lier. The greer he hdrogen ion onenrion, he more idi he soluion. Riher Sle (Seismi Aivi) n where is he inensi of he erhquke, nd n is how muh he erh moves on norml d ( miron = 0 m) R Deiels db (Sound nensi) db 0 o ws where is he inensi of he sound mesured in, nd o is he sofes udile meer ws sound 0. meer 5