Eponenial and Logarihmic Funcions -- ANSWERS -- Logarihms racice Diploma ANSWERS www.puremah.com
Logarihms Diploma Syle racice Eam Answers. C. D 9. A 7. C. A. C. B 8. D. D. C NR. 8 9. C 4. C NR. NR 6. 9. D. B. B. B. B 6. D 4. C. B. B NR. NR 4..4. C. B 7. B. C 4. B 8. C 6. C NR 7. 9.8 NR. 4. 7. B. C 9. A 8. D 6. B Logarihms racice Diploma ANSWERS www.puremah.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. 4) Change of Base in reverse log y = and y 6 log =. The answer is A..46 = w.46(4).8 = w lug in =.8 and = 4.78.8 = w Surac on oh sides.78.7 = w Cancel ou he negaives.78.7 = w.78.78.78 = ( w ).7 Isolae w y raising each side o he reciprocal eponen w =.8.46(4) (You can also solve his equaion y graphing y =.8 and y, hen find he -value of he = w poin of inersecion.) Logarihms racice Diploma ANSWERS www.puremah.com
) log ( y z) log( y z = log yz yz ) = log y z 6) 7 = + 4 4 4 7 Raise o reciprocal eponens [] 4 = ( + ) 4 7 = + Conver fracional eponen o a radical = 4 7 The answer is D. NR #) log + log y ( y) = log + log Facor ou he = log y Muliplicaion Law = log 8 We know y=8 = () Evaluae log 8 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! Logarihms racice Diploma ANSWERS 4 www.puremah.com
8) The iniial amoun A is. The final amoun A is 8. The lengh of ime is hours. The growh is ½. We wan o solve for. A = A 8 = 8 = = 4 = = = =. The half life is. hours, which is 6 minues. The answer is C NR #) a = a ( logc c ) + 8 + 8 a = ( alog c c) ower Law + 8 a = ( a) l ogc c = = + 8 Common Base = 8 = 4 The answer is 4.. 9) The poin (, a) can e ransformed o he poin (a, ) y drawing he inverse graph. The answer is A. Logarihms racice Diploma ANSWERS www.puremah.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 Logarihms racice Diploma ANSWERS 6 www.puremah.com
) Rewrie as: y = + 4 y Swap & y o ge: = + 4 Bring 4 o he lef side: 4= y Conver o log form (rememer a ase is always a ase) log 4 = y = ( ) Rewrie as f log 4 4) = 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 4) k T () = Te.4 6 = 8(.78).4.6 = (.78) =.4 minues Solve y graphing & poin of inersecion. Keep eponen in rackes! ) Graph 4 f = and log g = log 4 = in your calculaor, and noice he reflecion line is y = log 4 = ( ) + 6) Rewrie y g as y = g 4 + o see ha he graph has een shifed 4 unis righ. Since a logarihm graph has a verical asympoe along he y-ais, he asympoe is shifed 4 unis righ o make he line = 4. Thus, he domain is > 4 Logarihms racice Diploma ANSWERS 7 www.puremah.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. 8) 4 y = log 4 y = log ylog 4 = log log y = log4 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. Logarihms racice Diploma ANSWERS 8 www.puremah.com
) ( a) log 7 8 7 = 8a 7 a = 8 a = a = ( ) 4 4 4 a = = NR #) Deermine he eponenial regression equaion for he daa. Don use he acual years for he regression - use elapsed ime. For eample, can e replaced wih since one full year has passed. Year 4 rofi $8 $ $4 $49 $6 y = 49.6(.) Deermine he profi in y plugging in =, since you wan he enh year. The resuling value is $767., or 8 housand dollars. Answer = 8 NR #6) Graph y = 49.6(.) and y = 8. The -value of he poin of inersecion gives he value 8.. Since he 8 h year is under $8, he firs ime he usiness will e over $8 is he 9 h year. Answer = 9 ) Graph oh equaions and find he -value of he poin of inersecion. This occurs a. years, so he second usiness will overake he firs usiness during he h year. The answer is B Logarihms racice Diploma ANSWERS 9 www.puremah.com
) a4 = 4 4 a4 = a = 4 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 4) y = logc a = logc a = logc a c = a = a = a The answer is B NR #7) Group like erms log + log = 8 log = 8 8 log = 8 = = 9.8 ) 4 4 4 = = 6) loga + y = loga z y = loga z loga z y = loga 7) A = A A = 6 A = 88. 8) = (.) log = log (.) log = log + log(.) log = + log. log = log. log = log. The answer is D. The answer is 9.8 Logarihms racice Diploma ANSWERS www.puremah.com
9) log + log + = log ( )( + ) = ( )( + ) = log log 4 = = =± ) log 6 6 log 6 6 log 6 log 6 6 = = 8 The answer is D. ) log c log c a 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( + )( ) = = ( + )( ) = + = + = ( + 4)( ) = 4, Rejec -4 = Logarihms racice Diploma ANSWERS www.puremah.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 > Logarihms racice Diploma ANSWERS www.puremah.com
Wrien Response : Solving Graphically Graph y = 7 8 y = 4 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 8 = 4 4 ( ) ( ) = ( ) = 4 = 4 = 4 = = 4 =.7 4 8 Solve Using Logarihms 7 8 = 4 log 7 8 = log 4 log 7 + log8 = log 4 log 7 + ( ) log8 = log 4 log 7 + log8 log8 = log 4 log8+ log 4 = log8 log 7 ( log8+ log 4) = log8 log 7 log8 log 7 = log8+ log 4 =.7 Graph y = y = 4 log log Use a window of : [,, ] y: [-,, ] Answer: = 6 Logarihms racice Diploma ANSWERS www.puremah.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. = = 4.74 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 8 A = A = 8 =. years If he ligh inensiy is 64% of he iniial amoun, Surac he rae from he iniial amoun is A and he final amoun since i is a decreasing is.64a. Simplify and solve y graphing. percen. Simplify and solve A= A y graphing. A= A.64A = A 4 A = A(.97).64 = 4 = (.97) =. m =.77 years Logarihms racice Diploma ANSWERS 4 www.puremah.com