Properties of Logarithms. Solving Exponential and Logarithmic Equations. Properties of Logarithms. Properties of Logarithms. ( x)

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1 Properies of Logrihms Solving Eponenil nd Logrihmic Equions Properies of Logrihms Produc Rule ( ) log mn = log m + log n ( ) log = log + log Properies of Logrihms Quoien Rule log m = logm logn n log7 = log7 log7

2 Properies of Logrihms Power Rule log n m = nlog m log ( ) = log 4 Properies of Logrihms Chnge of Bse Rule log m log m = log g ln log = ln Properies of Logrihms log m = log n m = n ( ) ln + = ln + = 6

3 Properies of Logrihms log = log = log 0 = log = Emple Evlue: ( log )( ) log4 = = log 8 = (log + log 4) log ( 4) 8 Emple Epress s single log: log+ log+ log4 = log + log + log4 = log 4 = log 7 4 = log 6 ( ) 9

4 Emple Epnd erms: ln y z = ln + ln + ln z ln y = ln + ln + ln z ln y 0 Emple Collec erms: ln ln y =ln ln y = ln y = ln y Emple Condense log + log ( ) log ( ) 4 ( ) ( ) = + 4 log log log ( ) = + 4 log log log 4 ( ) ( ) = log log = log 4 4

5 Emple Wrie s he sum nd/or difference of logrihms. Epress ll powers s fcors log + ( ) + log = log log + ( ) ( ) ( ) = + + log log log = log+ log + log ( ) ( ) Solving Eponenil Equions Terms h hve se on one side nd power of h se on he oher side cn e solved using he propery: m n = m= n 4 Emple Solve = = 4 = + + = 4 0 ( )( ) = 0 =, =

6 Prcice Solve = + = + = + = 6 Solving Eponenil Equions Wch for qudric forms where we hve = ( ) We cn ofen fcor epressions conining erms like nd 7 Emple e Wrie in qudric form Fcor e 8= 0 ( ) e e 8= 0 ( e )( e + ) = 4 0 Use Zero Propery e 4 = 0 or e + = 0 Solve for e e = 4 or e = Recll R(e ) >0 Tke he ln of ech side e = 4 = ln 4 8 6

7 Solving Eponenil Equions Generl Guidelines Sep : Isole he eponenil epression Pu your eponenil epression on one side everyhing ouside of he eponenil epression on he oher side of your equion Sep : Tke he nurl log of oh sides The inverse operion of n eponenil epression is log. Mke sure h you do he sme hing o oh sides of your equion o keep hem equl o ech oher 9 Solving Eponenil Equions Sep : Use he properies of logs o pull he ou of he eponen Sep 4: Solve for Now h he vrile is ou of he eponen, solve for he vrile using inverse operions o complee he prolem 0 + Emple ( ) Isole he eponen Tke he log of ech side 0 = + ( 0) = + = log Solve for = log 7

8 Prcice e = 0 Tke he ln of ech side Solve for = ln0 = ln 0 Solving Logrihmic Equions of he Form log = y Sep : Wrie s one log isoled on one side Ge your log on one side everyhing ouside of he log on he oher side of your equion using inverse operions. Also use properies of logs o wrie i so h here is only one log Sep : Use he definiion of logrihms o wrie in eponenil form A reminder h he definiion of logrihms is he logrihmic funcion wih se, where > 0 nd 0, nd is defined s log = y if nd only if y = Solving Logrihmic Equions of he Form log = y Sep : Solve for Now h he vrile is ou of he log, solve for he vrile using inverse operions o complee he prolem Sep 4: Verify he domin This is necessry s he domin of log() is sricly posiive rels 4 8

9 Emple Use properies of log Rewrie s eponen Cross muliply Epnd Solve nd verify domin log ( + 4) log ( + ) = + 4 log = = = + 9 ( ) + 4 = = 9+ 8 = 4 Emple + ln = 4 Surc Propery of log ln = ln ( ) = ln = Muliply y ( ) Wrie in eponenil form = e Add Verify domin { e } = + e + e > Prcice Solve for log = log + ( ) ( ) = ( )( + ) ( )( + ) = log - + log + log0 = log - 0 = = = 0 ( + 4)( ) = 0 = 4, = ( ) ( ) disgrd negive soluion nd = 7 9

10 Solving Mied Equions Eponenil or logrihmic equions h mi ses + = Equions h mi eponenil nd logrihmic epressions + ln = Equions h mi rnscendenl nd lgeric epressions + = Use Grphic Mehod o solve hese mied equions 8 Emple + = 9 Emple + ln = 0 0

11 Prcice + = Newon's Lw of Cooling ( ) = f + ( 0 f ) T T T T e k T0 = emperure ime 0 T = finl emperure (mien) f T 0 T f k = Δ ln T T f = ime from T o T 0 TV Shows like CSI nd Numrs use funcions like he Logisic Funcions nd Newon s Lw of Cooling o solve crimes

12 Forensic Scieniss Forensic scieniss ke he emperure of ded ody les wice, ech mesuremen some ime pr crees d o erpole ck o when he deh 4 Our Prolem A murder vicim is found 8:0 AM nd h he emperure of he ody h ime is 0 o C. One hour ler i ws 8 o C, nd he room in which he murder vicim ly ws consn o C. The norml emperure of humn ody when i is live is 7 o C Wh ws he ime of deh? Our D Deh Discovered Ler Much Ler? 8:0 9:0 7 O C 0 O C 8 O C O C ime? 0 emperure T 0 = 0 T = 8 T f = 6

13 Our equion ecomes ( ) f ( 0 f ) T T T T e k = + = + ( 0 ) e k = + 8e k We mus find k T l T 0 f 0 4 k = ln T T = ln Δ = ln 8 f We hen hve ( ) = + 8e k T = + 8 = + 8 ln( 4) ( e ) ln( 4 ) e = + 8 ln( 4 ) ( e ) 7 I remins o find ou when he murder occurred A he ime of deh, he ody ws 7 o C So we need o solve for 7 = = 8 4 = 8 4 ln = ln = ln ln ( ) = 8.9 ln ( 4) 8 Beginning 8:0, we mus surc.9 hours o deermine he ime of deh 8: = :86 :8.6 6:9 The ime of deh is ou 6:9 AM