Exponential and Logarithmic Functions

Transcription

1 Chaper 6 Eponenial and Logarihmic Funcions 6.R Chaper Review. f () = = 5 = (5 )=+3 5 = = + 3 (5 )=+3 +3 = 5 f +3 ()= 5. f () = 3 + = 3 + = 3 + (3 + ) = 3 + = + = 3 ( +) = 3 = 3 + f () = 3 + Inverse Inverse Domain of f = range of f = all real numbers ecep 5 Range of f = domain of f = all real numbers ecep 5 Domain of f = range of f = all real numbers ecep 3 Range of f = domain of f = all real numbers ecep 643

2 Chaper 6 Eponenial and Logarihmic Funcions 3. f () = = = ( ) = = = + = + f () = + Inverse Domain of f = range of f = all real numbers ecep Range of f = domain of f = all real numbers ecep 0 4. f () = = = Inverse = 0 = + 0 f () = + 0 Domain of f = range of f = all real numbers greaer han or equal o Range of f = domain of f = all real numbers greaer han or equal o 0 5. f () = 3 3 Domain of f = range of f = all real numbers ecep 0 = 3 3 = 3 3 Inverse Range of f = domain of f = all real numbers ecep 0 3 = 3 3 = 3 = 7 3 f () =

3 Secion 6.R Chaper Review 6. f () = 3 + = 3 + = = = ( ) 3 f () = ( ) 3 Inverse Domain of f = range of f = all real numbers Range of f = domain of f = all real numbers 7. log ( 8) = log 3 = 3log = 3 8. log 3 8 = log = 4log 3 3 = 4 9. ln e = 0. e ln0. = 0.. log 0.4 = 0.4. log 3 = 3 log = 3 uv 3. log 3 w = log 3 uv log 3 w = log 3 u + log 3 v log 3 w = log 3 u + log 3 v log 3 w 4. log ( a b) 4 = 4log ( a b) = 4 log a + log b = 8log a + log b 5. log( 3 +) = log + log( 3 +) 6. log = log ( + 5 ) log 5 7. ln = ln + ( ) = 4 log a+ log b = log + log 3 ( + ) ( ) ( ) = log 5 ( +) log 5 ( ) ln( 3) = ln + ln + = ln + 3 ln + ( ) ln( 3) ln = ln 3+ = ln( + 3) ln( ) ln( ) ( ) 3 ln( 3) = ( ln( + 3) ln( )( ) ) ( ) = ln(+3) ln( ) ln( ) ( ) 9. 3log 4 + log 4 = log 4 ( ) 3 + log 4 = log = 5 log log log 3 = log 3 = log log 4 4 = log ( ) ( ) + log 3 = log 3 6 = log = log 3 + log

4 Chaper 6 Eponenial and Logarihmic Funcions. ln + ln + ln ( ) = ln + ln ( ) = ln + = ln + ( )( + ) = ln ( +) = ln( +) = ln( +). log( 9) log( ) = log ( 3)( + 3) ( + 3)( + 4) = log log+3log [ log( + 3) + log( ) ] = log +log 3 log( + 3)( ) 4 3 = log4 3 log (( + 3)( ) ) = log (( + 3)( ) ) 4. ln + ( ) 4ln = ln + [ ln( 4) + ln ] = ln + ( ) ln 4 ( ) ln ( +) ln( ( 4) ) = ln 4 ( ) 6 ( 4) ln ( ( 4) ) 5. log 4 9 = log9 log4 =.4 6. log log = log = ln = + ln C ln = ln e + ln C ln = ln Ce = Ce ( ) 9. ln( 3) + ln( +3) = + C ln( 3)( + 3) = + C ( 3)( + 3) = e + C 9 = e + C = 9 + e +C = 9 + e + C 8. ln( 3) = ln( ) + lnc ln( 3) = ln C ( ) 3 = C = C ln( ) + ln( +) = + C ln( )( +) = + C ( )( +) = e +C = e +C = + e + C +C = + e 3. e + C = + 4 ln e + C = ln( + 4) + C = ln( + 4) = ln( + 4) C 3. e 3 C = ( + 4) ln e 3 C = ln( + 4) 3 C = ln( + 4) 3 = ln(+4)+c = ln(+4)+c 3 646

5 Secion 6.R Chaper Review 33. f () = 3 Using he graph of =, shif he graph 3 unis o he righ. Domain: (, ) Range: (0, ) Horizonal Asmpoe: = f () = + 3 Using he graph of =, reflec he graph abou he -ais, and shif vericall 3 unis up. Domain: (, ) Range: (,3) Horizonal Asmpoe: = f () = 3 Using he graph of = 3, reflec he graph abou he -ais, and shrink vericall b a facor of. Domain: (, ) Range: (0, ) Horizonal Asmpoe: = f () = + 3 Using he graph of = 3, shrink he graph horizonall b a facor of, and shif vericall uni up.. Domain: (, ) Range: (, ) Horizonal Asmpoe: = 647

6 Chaper 6 Eponenial and Logarihmic Funcions 37. f () = e Using he graph of = e, reflec abou he -ais, and shif up uni. Domain: (, ) Range: (,) Horizonal Asmpoe: = 38. f () = 3e Using he graph of = e, srech vericall b a facor of 3. Domain: (, ) Range: (0, ) Horizonal Asmpoe: = f () = 3 + ln Using he graph of = ln, shif he graph up 3 unis. Domain: (0, ) Range: (, ) Verical Asmpoe: = f () = ln Using he graph of = ln, shrink vericall b a facor of. Domain: (0, ) Range: (, ) Verical Asmpoe: = 0 648

7 4. f () = 3 e Using he graph of = e, reflec he graph abou he -ais, reflec abou he -ais, and shif up 3 unis. Domain: (, ) Range: (,3) Horizonal Asmpoe: = 3 Secion 6.R Chaper Review 4. f () = 4 ln( ) Using he graph of = ln, reflec he graph abou he -ais, reflec abou he - ais, and shif up 4 unis. Domain: (,0) Range: (, ) Verical Asmpoe: = = ( ) = 4 = 4 = 4 = = = 4 ( 3 ) 6+ 3 = 8+ 9 = = 9 = 6 = = = 3 + = + = 0 = ± 4 4()( ) () = ± 4 = ± 3 4 = ± = = 3 or = + 3 ( ) = = = =0 = ( ) ± = 3 or = ()( ) () = ± 4 = ± 3 4 = ± 3 649

8 Chaper log 64 = 3 3 = = 64 3 ( ) = 3 64 Eponenial and Logarihmic Funcions = log = 6 = ( ) 6 = ( ) 6 = 3 = = 3 + log( 5 ) = log( 3 + ) log5=(+)log3 log5= log3 + log3 log5 log3 = log3 (log5 log3) = log3 log3 = log5 log = ( 3 ) = 3 3 ( ) = = 9 5 = = log 3 = = 3 = 9 = 8 = = = ( ) 5 3 = +5 3 = = = ( )( + 3) = or = = 7 log( 5 + ) = log( 7 ) ( + )log5=( )log7 log5 + log5=log7 log7 log5 log7 = log7 log5 (log5 log7) = log7 log5 = log7 log5 log5 log = 5 ( 5 ) = = 5 4 = 4 = 0 ( 6)( + ) = 0 = 6 or = = ( ) = + 3 = + = + = = = = 0 ( ) = ln0 ln 5 ln +ln5=ln0 ln + ln5= ln0 (ln ln0) = ln5 = ln5 ln ln0 = 650

9 Secion 6.R Chaper Review 57. log 6 ( + 3) + log 6 ( + 4)= log 6 ( + 3)( + 4) = ( + 3)( + 4) = = = 0 ( + 6)( +) = 0 = 6 or = The logarihms are undefined when = 6, so = is he onl soluion. 58. log 0 (7 )= log 0 log 0 (7 )= log 0 7 = 7 + = 0 ( 4)( 3) = 0 = 4 or = e = 5 = ln5 = +ln5 = ln = 3 ln 3 = ln3 + 3 ln=(+)ln3 3ln= ln3 + ln3 3 ln ln3 = ln3 (3ln ln3)= ln3 = ln3 3ln ln e = 4 = ln4 = +ln4 = ln = 3 ln 3 = ln3 3 ln= ln3 3 ln ln3 = 0 ( ln ln3) = 0 = 0 or = ln3 ln = 0 or h(300)= ( 30(0)+ 8000)log = 8000log.53333= 39.5 meers 64. h(500)= ( 30(5) )log = 850log.5 = 48 meers 65

10 Chaper 6 Eponenial and Logarihmic Funcions 65. h( ) = = ( 30( 00) )log = ( 5000)log 760 = log =0 log = 760 = 760 = 7.6 mm P = 5e 0. d (a) P = 5e 0.(4) = 5e 0.4 = 37.3 was 68. L = logd (a) L = log h( ) = = ( 30(5) )log = ( 850)log = log log 760 = = = 760 = mm 8900 (b) 50 = 5e 0. d = e 0.d ln=0.d d = ln = 6.9 decibels 0. (b) 4 = log d 5 = 5.log d log d = d = inches 69. (a) P = % (b) P = (c) as, P = = 90% 4 (d) 40 = = = 3 4 ln 5 8 = ln % ln 5 8 = ln 3 ln = ln 3.63 monhs 4 65

11 Secion 6.R Chaper Review (e) 70 = = = 3 4 ln 0.5 ( ) = ln 3 4 ( ) = ln 3 4 ln 0.5 ( ) ln 0.5 = ln monhs 70. m = ln( ) = ln( 5000) 4. monhs 7. (a) n = (b) log0000 log90000 log( 0.0) = 9.85 ears n = log0.5i log i log( 0.5) 0.5i log = i log0.85 = log0.5 log0.85 = 4.7 ears 7. A = ()(8) =0000(.0) 36 = $0, P = A + r n n = (8) = $4, (a) 5000 = 60.7e r (0) = e 0 r ln8.063= 0r r = ln % 0 (b) A = 4000e (0) = $3,49.4 The banks claim is correc. 75. L 0 4 ( ) =0log = ( 0log08 )=0 8 = 80 decibels 653

12 Chaper 6 Eponenial and Logarihmic Funcions 76. Chicago: M() = log 3 = log( 0 3 ) = 3.0 log( ) + log( 0 3 ) = 3 log( ) + 3 = 3 log( ) = 0 0 log( ) =0 0 = 0 0 = 77. A = A 0 e k A 0 =A 0 ek(5600) 0.5 = e 5600 k ln0.5=5600k k = ln San Francisco: M() = log 3 = log( 0 3 ) = 6.9 log( ) + log( 0 3 ) = 6.9 log( ) + 3 = 6.9 log( ) = log( ) =0 3.9 = A 0 = A 0 e = e ln0.05= ln0.05 = 4,59 ears ago Using u = T + (u 0 T)e k where = 5, T = 70, u 0 = 450, u = 400: 400 = 70 + (450 70)e k (5) 330 = 380e 5k = e 5k 5k = ln k = ln Find ime for emperaure of 50 F: 50 = 70 + (450 70)e = 380e = e ln0.056= = ln minues

13 Secion 6.R Chaper Review 79. (a) Graphing: (c) N =000e ( 7) 34 baceria (d) and (e) Graphing: (b) =000( ) = e ln =000( e ln ) ( =000e ln ) N = N o e ,N 0 = 000,k = (a) Graphing: (c) A =004e ( ) $ (d) and (e) Graphing: (b) =004(.057).057 = e ln.057 =004 e ln.057 ( ) =000e ( ) A = A 0 e ,A 0 = 004,k = P = P 0 e k =5,840,445,6 e 0.033(3) = 6,078,90, A = A 0 e k ek (5.7) A =A 0 0 = e5.7 k ln0.5=5.7k k = ln (0) In 0 ears: A = 00e In 40 ears: A = 00e 0.353(40) = 7. grams = 0.5 grams 655

14 Chaper 6 Eponenial and Logarihmic Funcions 83. (a) 0.8 P(0)= +.67e = (0) +.67 = (b) 0.8 (c) Graphing: (d) Using INTERSECT we have: % use Windows 98 in