Eponenial and Logaihmic Equaions and Popeies of Logaihms Popeies Eponenial a a s = a +s a /a s = a -s (a ) s = a s a b = (ab) Logaihmic log s = log + logs log/s = log - logs log s = s log log a b = loga + logb 2 Popeies Eponenial Logaihmic a = a y = y log = logy = y a = a log b b = a 0 = log b = 0 a - = /a log b b log a - = - loga log b b
Popeies Base 0 log0 log 0 log0 0 log Base e ln e ln 0 ln e e ln Remaks We mus lean how o use hese popeies o solve logaihmic and eponenial equaions epand logaihmic epessions collec logaihmic epessions 5 Using log/s = log - logs Epand log 5 log log5 Using log s = log + logs log log5 log Using log s = s log log log5 log 6 2
Rewie as a single logaihm log log log log Using log s = s log log Using log s = log + logs 7 Change of Base Fomula If b 0, b, a 0, a, and 0, logb hen loga log a b Convesion o base 0 log loga log a Convesion o base e ln loga ln a 8 Evaluae log 8 o decimal places 0.96 9
Solving Eponenial Equaions Using logaihmic Foms. Rewie he equaion wih he em conaining he eponen by iself on one side 2. Divide boh sides by he coefficien of he em conaining he eponen. Change he new equaion o logaihmic fom. Solve fo he vaiable 0 Solve 5000 20 0 o 5 decimal places 5000 20 2500 0 log 2500.979 Solve 5000 20 0 o 5 decimal places 2
Solve 5000 20 0 Gaphing Zeo Mehod o 5 decimal places Solve 5000 20 0 o 5 decimal places Solving Eponenial Equaions Using logaihmic Popeies. Rewie he equaion wih a base aised o a powe on one side 2. Take he logaihm, base e o 0, of boh sides. Use logaihmic popey o emove he vaiable fom he eponen. Solve fo he vaiable 5 5
50 Solve 50 0 o decimal places ln ln50 ln ln50 ln50 ln ln50 ln 5.56 6 Solve 50 0 Gaphing Zeo Mehod o decimal places 7 Solve 50 0 o decimal places 8 6
Remaks Fo eponenial gowh we use base wo when we know D he ime o double A O 2 D Fo eponenial decay we use base one-half when we know H he ime o half H AO AO 2 2 H 9 Remaks The following is a discussion of solving vaious inees ae fomulas These ae fomulas we will sudy in his chape 20 Remaks We can use popeies of log o solve ou gowh and decay equaions fo one of he vaiables Suppose we ake A P ln A ln P ln A ln P ln and wan o solve fo ln A ln P ln ln Aln P ln ln A ln P ln 2 7
Suppose a young couple have $000, bu need $800 o puchase an engagemen ing How long do hey have o wai if hey have accoun eaning 5% inees monhly ln A ln P ln 800 ln 000 ln.05 ln 2 monhs O 9 yeas and 5 monhs 22 TI Finance App N Numbe of oal paymens. (n*) I% Annual Pecenage Rae as a %, no a decimal PV Pesen Value: Saing balance of a loan, o savings accoun PMT Recuing paymen FV Fuue value/accued amoun. (0 fo loans) P/Y Paymens pe yea (mos ofen 2 fo monhly paymens) C/Y Compounding cycles pe yea (n; ofen maches P/Y) PMT:END BEGIN Time when paymens ae made (almos always END unless specified) 2 Suppose a young couple have $000, bu need $800 o puchase an engagemen ing How long do hey have o wai if hey have accoun eaning 5% inees monhly O 9 yeas and 5 monhs 2 8
Remaks We can use popeies of log o solve ou gowh and decay equaions fo one of he vaiables Suppose we ake A P ln A ln P ln A ln P ln ln A ln P ln ln Aln P ln and wan o solve fo ln A ln P ln e ln Aln P e ln Aln P 25 Remaks We can use popeies of eponens o solve ou gowh and decay equaions fo one of he vaiables Suppose we ake A P A P A P A P and wan o solve fo 26 Suppose a young couple have $000, bu need $800 o puchase an engagemen ing How much annual inees do hey need o ean if hey do no wan o wai moe han yeas e ln Aln P A P e 000 800 ln 800ln000 0.25 0.25 O 2.5% 27 9
Suppose a young couple have $000, bu need $800 o puchase an engagemen ing How much annual inees do hey need o ean if hey do no wan o wai moe han yeas O 2.5% 28 Remaks We can use popeies of log o solve ou gowh and decay equaions fo one of he vaiables Suppose we ake A P ln A ln P ln A ln P ln ln A ln P ln ln Aln P ln and wan o solve fo ln A ln P ln e ln Aln P e ln Aln P 29 Conveing Tems We can conve fom peiodic o coninuous aes using R P Ae and P A n n R e = n R ln n n n nln n R nln n 0 0
Find he equivalen coninuous ae o.25% compounded semi-annually R nln n 0.025 R 2ln 2 0.02 O.2% Conveing Tems We can conve fom one peiod o anohe peiod N R using P A and P A N n N R N n R N n nn n nn n R N n nn R N N n nn R N N n 2 Find he equivalen monhly ae o.25% compounded semi-annually nn R N N n 2 2 0.025 R 2 2 2 0.02 O.2%