Logarihms racice Eam - ANSWERS Answers. C. D 9. A 9. D. A. C. B. B. D. C. B. B. C NR.. C. B. B. B. B 6. D. C NR. 9. NR. NR... C 7. B. C. B. C 6. C 6. C NR.. 7. B 7. D 9. A. D. C Each muliple choice & numeric response is worh mark (Toal = marks) Each wrien response is worh 6 marks (Toal = marks) To deermine your score on he eam, add your marks and divide y. rinciples of Mah : Logarihms racice Eam - ANSWERS www.mah.com
) Graphical Soluion: The quesion ells you ha >, so you could graph y = and y = o see wha each case would look like. These graphs are symmerical wih respec o he y ais ( = ). = = Algeraic soluion: ( ) The answer is C. Compared o he original of, i s refleced in he y-ais. ) To graph y = log, change of ase is required since he calculaor only acceps ase logs. In your calculaor, you would use ) log log = log Change of Base log-log = Division Law log-log -log = Since log= -log log = Cancel ou he negaives log = log The answer is D. ).6 = w Change of Base in reverse log y = and y 6 log =. The answer is A..6(). = w lug in =. and =.7. = w.7.7 = w.7.7 = w.7 = ( w ).7.7 Surac on oh sides Cancel ou he negaives.7 Isolae w y raising each side o he reciprocal eponen w =..6() (You can also solve his equaion y graphing y =. and y, hen find he -value of he = w poin of inersecion.) rinciples of Mah : Logarihms racice Eam - ANSWERS www.mah.com
) log ( y z) log( y z = log yz yz ) = log y z 6) 7 = + 7 Raise o reciprocal eponens [] = ( + ) 7 = + Conver fracional eponen o a radical = 7 The answer is D. NR #) log + log y ( y) = log + log Facor ou he = log y Muliplicaion Law = log We know y= = () Evaluae log using change of ase = The answer is. -- 7) Graph y = and y = in your calculaor and find he -value of he poin of inersecion. Rememer o keep your eponen in rackes! rinciples of Mah : Logarihms racice Eam - ANSWERS www.mah.com
) The iniial amoun A is. The final amoun A is. The lengh of ime is hours. The growh is ½. We wan o solve for. A = A = = = = = = =. The half life is. hours, which is 6 minues. The answer is C NR #) a = a ( logc c ) + + a = ( alog c c) ower Law + a = ( a) l ogc c = = + Common Base = = The answer is.. 9) The poin (, a) can e ransformed o he poin (a, ) y drawing he inverse graph. The answer is A. rinciples of Mah : Logarihms racice Eam - ANSWERS www.mah.com
) Use he formula A = A A = he fuure score S A = he iniial score = elapsed days d = rae. Decreasing percenage; surac his decimal from. (.97) = he percenage loss is per day, so he period is. lug hese ino he formula o ge S = (.97) d The answer is D. ) db db db ( I ) ( I ) db = log db = log = I I = db db db db I = Common denominaor: = = I = The answer is C ) db I = I = I = I = I = NR #) log log ( ) log = The answer is rinciples of Mah : Logarihms racice Eam - ANSWERS www.mah.com
) Rewrie as: y = + y Swap & y o ge: = + Bring o he lef side: = y Conver o log form (rememer a ase is always a ase) log = y = ( ) Rewrie as f log ) = y y log = log Solve for y y aking he log of oh sides log = ylog log y = log y = log Change of ase in reverse The negaive in fron indicaes a reflecion in he -ais. NR ) k T () = Te. 6 = (.7)..6 = (.7) =. minues Solve y graphing & poin of inersecion. Keep eponen in rackes! ) Graph f = and log g = log = in your calculaor, and noice he reflecion line is y = log = ( ) + 6) Rewrie y g as y = g + o see ha he graph has een shifed unis righ. Since a logarihm graph has a verical asympoe along he y-ais, he asympoe is shifed unis righ o make he line =. Thus, he domain is > rinciples of Mah : Logarihms racice Eam - ANSWERS 6 www.mah.com
7) If you are solving wo equaions graphically, he -value of he poin of inersecion is wha you require. B is he incorrec procedure. ) y = log y = log ylog = log log y = log Graphing his epression (rememer o keep he denominaor in rackes) gives graph D. 9) + f = 7a 7 + = a + = 7a + = a 7 + log = log a 7 log log 7 = ( + ) log a log log 7 = + log a log log 7 = log a The answer is A. rinciples of Mah : Logarihms racice Eam - ANSWERS 7 www.mah.com
) ( a) log 7 7 = a 7 a = a = a = ( ) a = = ) a = a = a = The answer is B ) log 6 Rewrie as log ( 6 ) log The numeraor is defined for < 6 The denominaor is defined for > The enire graph is defined eween and 6, wih he ecepion of = since ha makes he denominaor zero. The answer is C ) y = logc a = logc a = logc a c = a = a = a The answer is B ) Divide y hree o evaluae one-hird = = ) loga + y = loga z y = loga z loga z y = loga 6) A = A A = 6 A =. 7) = (.) log = log (.) log = log + log(.) log = + log. log = log. log = log. The answer is D. rinciples of Mah : Logarihms racice Eam - ANSWERS www.mah.com
NR #) Group like erms log + log = log = log = = = 9. The answer is 9. ) log + log + = log ( )( + ) = ( )( + ) = log log = = =± 9) log 6 6 log 6 6 log 6 log 6 6 = = The answer is D. ) a log c log c a log c+ log c = a log c = a log c = ( a ) ) The graph has een moved down y hree unis. There is a horizonal asympoe a he line y = -, and he graph is aove his line. The range is y > - ) ( + ) + ( ) log log = log( + )( ) = = ( + )( ) = + = + = ( + )( ) =, Rejec - = rinciples of Mah : Logarihms racice Eam - ANSWERS 9 www.mah.com
Wrien Response : Domain > - Range y ε R Equaion of Asympoe = - -inercep (-, ) y-inercep (,.) y-value when =.6 log ( + ) Horizonal ranslaion of unis lef. The domain of a logarihmic epression can e found y seing wha is in he rackes greaer han zero. For he epression alog( + c) + d + c > > c c > rinciples of Mah : Logarihms racice Eam - ANSWERS www.mah.com
Wrien Response : Solving Graphically Graph y = 7 y = A window seing ha will le you see he poin of inersecion clearly is : [-,,.] y: [-.,.,.] The answer is =.7 Solve Using a Common Base 7 = ( ) ( ) = ( ) = = = = = =.7 Solve Using Logarihms 7 = log 7 = log log 7 + log = log log 7 + ( ) log = log log 7 + log log = log log+ log = log log 7 ( log+ log ) = log log 7 log log 7 = log+ log =.7 Graph y = y = log log Use a window of : [,, ] y: [-,, ] Answer: = 6 rinciples of Mah : Logarihms racice Eam - ANSWERS www.mah.com
Solve for A= A A A = A log = log A log A log A = log log A log A log = ( ) = ( log A log A ) log Wrien Response : lug in your values and solve y graphing. (Or use he equaion derived aove.) A= A 9 = 6. = =.7 hours If he populaion of he own doules, he iniial amoun is A and he final amoun is A. Simplify and solve y graphing. A= A A = A = =. years If he ligh inensiy is 6% of he iniial amoun, Surac he rae from he iniial amoun is A and he final amoun since i is a decreasing is.6a. Simplify and solve y graphing. percen. Simplify and solve A= A y graphing. A= A.6A = A A = A(.97).6 = = (.97) =. m =.77 years rinciples of Mah : Logarihms racice Eam - ANSWERS www.mah.com