4.5 COLLEGE ALGEBRA 11/5/2015. Property of Logarithms. Solution. If x > 0, y > 0, a > 0, and a 1, then. x = y if and only if log a x = log a y.

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1 /5/05 0 TH EDITION COLLEGE ALGEBRA 4.5 Eponntial and Logarithmic Equations Eponntial Equations Logarithmic Equations Applications and Modling LIAL HORNSBY SCHNEIDER Proprty of Logarithms If > 0, y > 0, a > 0, and a, thn = y if and only if log a = log a y. Eampl Solv 7 =. Giv th solution to th narst Th proprtis of ponnts givn in Sction 4. cannot b usd to solv this quation, so w apply th prcding proprty of logarithms. Whil any appropriat bas b can b usd, th bst practical bas is bas 0 or bas. W choos bas (natural) logarithms hr Eampl Solv 7 =. Giv th solution to th narst 7 I n 7 In Proprty of logarithms In 7 In Powr of logarithms In In 7 Divid by In Us a calculator. Th solution st is {.77}. Caution B carful whn valuating a In quotint lik in Eampl. Do not In 7 confus this quotint with In, which can 7 b writtn as In In 7. W cannot chang th quotint of two logarithms to a diffrnc of logarithms. In In In

2 /5/05 Eampl Solv 3 =.4 +. Giv th solution to th narst 3.4 In 3 In.4 Tak natural logarithms on both sids. ( ) In 3 ( ) In.4 Proprty powr In 3 In 3 In.4 In.4 Distributiv proprty Eampl Solv 3 =.4 +. Giv th solution to th narst In 3 In.4 In.4 In 3 Writ th trms with on on sid ( In 3 In.4) In.4 In 3 Factor out. In.4 3 In 3.4 In.4 In 3 In 3 In.4 Divid by In 3 In.4. Powr proprty Eampl Solv 3 =.4 +. Giv th solution to th narst This is act. In. In 3 In 9 In.4 In.48 9 In.4.3 Th solution st is {.3}. Apply th ponnts. Product proprty; Quotint proprty This is approimat Eampl 3 SOLVING BASE EXPONENTIAL S Solv th quation. Giv solutions to th narst a In In 00 In 00 In = Tak natural logarithms on both sids Eampl 3 SOLVING BASE EXPONENTIAL S Solv th quation. Giv solutions to th narst a. 00 In Rmmbr both roots. Th solution st is {.30}. Squar root proprty Us a calculator Eampl 3 SOLVING BASE EXPONENTIAL S Solv th quation. Giv solutions to th narst 3 b m n m n a a a m a m n 3 Divid by ; a. n a Tak natural logarithms on both sids. In In 3 In In 3 Powr proprty 4.5 -

3 /5/05 Eampl 3 SOLVING BASE EXPONENTIAL S Solv th quation. Giv solutions to th narst 3 b. 4 In 3 In = In 3 Multiply by ½.549 Th solution st is {.549} Eampl 4 SOLVING A LOGARITHMIC Solv log( + ) log( + ) = log. log( ) log( ) log log log Quotint proprty ( ) Proprty of logarithms Eampl 4 SOLVING A LOGARITHMIC Solv log( + ) log( + ) = log. 0 Distributiv proprty Standard form ( 3)( ) 0 Factor. 3 or Zro-factor proprty Th proposd ngativ solution ( = 3) is not in th domain of th log in th original quation, so th only valid solution is th positiv numbr, giving th solution st {} Caution Rcall that th domain of y = log a is (0, ). For this rason, it is always ncssary to chck that proposd solutions of a logarithmic quation rsult in logarithms of positiv numbrs in th original quation Eampl 5 SOLVING A LOGARITHMIC Solv log(3 + ) + log( ) =. Giv th act valu(s) of th solution(s). log(3 ) log( ) log(3 ) log( ) log0 Substitut. log[(3 )( )] log0 Product proprty (3 )( ) 0 Proprty of logarithms Eampl 5 SOLVING A LOGARITMIC Solv (3 + ) + log( ) =. Giv th act valu(s) of th solution(s). 3 0 Multiply. 3 0 Subtract Quadratic formula

4 /5/05 Eampl 5 SOLVING A LOGARITMIC Solv (3 + ) + log( ) =. Giv th act valu(s) of th solution(s). 45 Th numbr is ngativ, so is ngativ. Thrfor, log( ) is not dfind and this proposd solution must b discardd. 45 Sinc >, both 3 + and ar positiv and th solution st is 45. Not Th dfinition of logarithm could hav bn usd in Eampl 5 by first writing log(3 ) log( ) log [(3 )( )] Product proprty 0 (3 )( ) 0, Dfinition of logarithm and thn continuing as shown abov Eampl SOLVING A BASE LOGARITHMIC Solv In In In( 3) = In. Giv th act valu(s) of th solution(s). In In In( 3) In In In( 3) In In In 3 3 In Quotint proprty Proprty of logarithms Eampl SOLVING A BASE LOGARITHMIC Solv In In In( 3) = In. Giv th act valu(s) of th solution(s). Multiply by 3. Solv for. Vrify that th solution st is {} Solving Eponntial Or Logarithmic Equations To solv an ponntial or logarithmic quation, chang th givn quation into on of th following forms, whr a and b ar ral numbrs, a > 0 and a, and follow th guidlins.. a () = b Solv by taking logarithms on both sids.. Log a () = b Solv by changing to ponntial form a b = (). Solving Eponntial Or Logarithmic Equations 3. log a () = log a g() Th givn quation is quivalnt to th quation () = g(). Solv algbraically. 4. In a mor complicatd quation, such as th on in Eampl 3(b), it may b ncssary to first solv for a () or log a () and thn solv th rsulting quation using on of th mthods givn abov. 5. Chck that th proposd solution is in th domain

5 /5/05 Eampl 7 TO THE STRENGTH OF A Th strngth of a habit is a function of th numbr of tims th habit is rpatd. If N is th numbr of rptitions and H is th strngth of th habit, thn, according to psychologist C. L. Hull, H 000( ), Eampl 7 TO THE STRENGTH OF A First solv th quation for kn. H 000( ) H 000 Divid by 000. whr k is a constant. Solv this quation for k H H Subtract. Multiply by and rwrit Eampl 7 TO THE STRENGTH OF A Now w solv for k. H In In 000 H In 000 H k In N 000 Tak natural logarithms on both sids. In = N Multiply by. Eampl 7 TO THE STRENGTH OF A H k In N 000 N Multiply by. With th final quation, if on pair of valus for H and N is known, k can b found, and th quation can thn b usd to find ithr H or N for givn valus of th othr variabl MODELING COAL CONSUMPTION Eampl 8 IN THE U.S. Th tabl givs U.S. coal consumption (in quadrillions of British thrmal units, or quads) for svral yars. Th data can b modld with th function dfind by f ( t).97 In t 0.4, t 80, Yar Coal Consumption (in quads) whr t is th numbr of yars aftr 900, and (t) is in quads MODELING COAL CONSUMPTION Eampl 8 IN THE U.S. a. Approimatly what amount of coal was consumd in th Unitd Stats in 003? How dos this figur compar to th actual figur of.3 quads? Th yar 003 is rprsntd by t = = 03. f ( 03).97 In Us a calculator. Basd on this modl,.54 quads wr usd in 003. This figur is vry clos to th actual amount of.3 quads

6 /5/05 MODELING COAL CONSUMPTION Eampl 8 IN THE U.S. b. Lt (t) = 5, and solv for t In t In t 7.4 In t t Add 0.4 Divid by.97; rwrit. Writ in ponntial form. t 3 Us a calculator. Add 3 to 900 to gt 03. Annual consumption will rach 5 quads in approimatly