ࡱ> 9 Vbjbj " gP*lllll\V\V\V8V(X,`Y`Y4YYYKKK-//////$ lSK6KKKSollYYkVhoooKltYY-oK-ono0 YTY 4;tP\V$ $~0$,x, ollll1. LOGARITMES 1.1. Definici   log a N = x ( ====== ( a x = N aix: 3 2 = 9 ( log 3 9 = 2 per tant: - log10 1000 = x per com que 10 x = 1000 llavors... x = 3 perqu 103 = 1000 - log4 2 = x per com que 4 x = 2 llavors... x = perqu 41/2 = 2 - log1/2 0,125= x per com que () x = 0,125 llavors... x = 3 perqu ()3 = 0,125 i tamb, - log5 239 = x per com que 5 x = 239 llavors... x = 3,39.. perqu 53.39.. = 239 tal com es veu, poden existir diferents bases, i per tant, diferents tipus de logaritmes, per solament es treballa amb dos tipus: - DECIMALS (o de Briggs) base 10 - NEPERIANS (o Naturals) base e = 2,7182.. 1.2. Consideracions  I loga 1 = 0 perqu ao = 1 log4 1 = 0 14o = 1 II loga a = 1 perqu a1 = a log5 5 = 1 51 = 5 III loga (am) = m perqu am = am log10 (103) = 3 103 = (10 3)  IV loga (-N) = ( perqu a x ( (-N) no pot haver cap nmero positiu que elevat a qualsevol nombre positiu o negatiu doni < 0 V loga 0 = - ( perqu a-( = 0 log2 0,000...001 = - 19 2-19 = 0,000....001 (a > 1) VI loga 0 = ( perqu a( = 0 log(1/2) 0,000...001 = 20 (1/2)-20 = 0,000...001 (a < 1) 1.3. Consideracions avanades   VI si a > 1 i a) N > 1 loga N = (+) log3 27 = x (( 3x = 27 ( x = 3 33 = 27 per tant, log3 27 = 3  b) N < 1 loga N = (-) log2 0,125 = x (( 2x = 0,125 ( x = -3 2-3 = 1/33 = 1/8 = 0,125 per tant, log2 0,125 = - 3  si a < 1 i a) N > 1 loga N = (-) log(1/5) 625 = x (( (1/5)x = 625 ( x = -4 (1/5)-4 = 5 4 = 625 per tant, log(1/5) 625 = - 4  b) N < 1 loga N = (+) log(1/2) 0,25 = x (( (1/2)x = 0,25 ( x = 2 (1/2)2 = 1/4 = 0,25 per tant, log(1/2) 0,25 = 2 Exercicis A) Calcular els segents logaritmes B) Expressar de forma logartmica 1. log 3 9 = 1. 43 = 64 ( 2. log 4 2 = 2. 7-2 = 1/49 ( 3. log 2 8 = 3. (1/2) 4 = 1/16 ( 4. log 5 1 = 4. (1/5) -3 = 125 ( 5. log 10 10.000 = 6. log 27 3 = C) Expressar de forma exponencial 7. log 2 0,25 = 1. log 2 1024 = 10 ( 8. log 10 0,001 = 2. log 7 1/49 = -2 ( 9. log 1/2 4 = 3. log 5 5 = 1 ( 10. log 3 81 = 4. log 1/3 243 = -5 ( 11. log 5 (1/3125) = 12. log 4 4096 = D) Calcular el valor de x 1. log x 1 = 0 2. log x 512 = 3 3. log x (1/3125) = -5 4. log 4 X = 2 5. log 9 X = 1/2 6. log 27 X = -1/3 2. TIPUS DE LOGARITMES   2.1. DECIMALS (Briggs) 2.2. NEPERIANS (Naturals) La base a = 10 La base a = e e= 2,7182... Nomenclatura  Log 10 N ( (Forma descriptura) Log e N (   log N = x ( ( 10 x = N ln N = x ( ( e x = N Nomenclatura (operativa: taules - mquines) si log 1 = 0 100 = 1 si ln 1 = 0 e 0 = 1 log 10 = 1 101 = 10 ln 7,389..= 2 e 2 = 7,389.. log 100 = 2 102 = 100 ln 10 = 2,302.. e 2,302.. = 10 ln 100 = 4,605.. e 4,605.. = 100 log 1000 = 3 103 = 1000 ln 200 = 5,298.. e 5,298.. = 200 ln 500 = 6,214.. e 6,214.. = 500 ............................................. ..................................................... log 256 = 2 ( 3 102(3 = 256 ln 256 = 5 ( 6 e5(6 = 256 log 256 = 2 , 40824.. ln 256 = 5 , 54517..  part sencera (2) : Caracterstica part sencera (5) : Caracterstica part decimal (40824..) Mantisa part decimal (54517..) Mantisa ( Important: En les logaritmes decimals si un nombre el multipliquem per una potncia de 10 la mantisa no varia i la caracterstica va augmentant en una unitat (en les logaritmes neperians passaria el mateix per amb potncies de e) log 3,56 = 0,551.. log 0,356 = -0,448.. ln 3,56 = 1,269.. log 35,6 = 1,551.. log 0,0356 = -1,448.. ln 3,56 x e = 2,269.. log 356 = 2,551.. log 0,00356 = -2,448.. ln 3,56 x e2 = 3,269.. log 3560 = 3,551.. log 0,000356 = -3,448.. precisament, 1- 551.. = 448.. 3. ANTI-LOGARITME  log 256 = 2,40824.. ln 256 = 5,54517..  ant-log (2,40824..) = 256 ant-ln (5.545..) = 256 10 2,40824.. = 256 e 5,54517.. = 256 3.1. COLOGARITME Cologaritme dun nmero (N) es el logaritme del seu invers (1/N)   Colog 256 = log (1 / 256) = - 2,40824.. Coln 256 = ln (1 / 256) = - 5,54517.. 3.2 CALCULADORA Calcular el logaritme de 27,931 pantalla tecla pantalla pantalla tecla pantalla  27.931 1,4460.. 7.89 2,065.. log 27,931 = 1,4460... ln 7,89 = 2,065.. Calcular lantilogaritme de 1,45678 pantalla tecla dinversa pantalla pantalla tecla dinversa pantalla  1.45678 28.627.. 1.45678 4,2921.. ant-log 1,45678 = 28,627... ln 1,45678 = 4,2921.. Exercicis: 1. Al multiplicar un nombre per 10.000 varia la mantisa del seu log? 2. Al multiplicar un nombre per 1000, que li passa a la seva caracterstica del seu log 3. Si el log 282 = 2,4502 calcular sense calculadora el log 2,82 = 4. Si el log 282 = 2,4502 calcular sense calculadora el log 0,00282 = 5. Si l ant-log (2,356) = 226,98.. calcular sense calculadora l ant-log (5,356) = 6. Si l ant-log (- 1,221) = 0,0601.. calcular sense calculadora l ant-log (-3,221) = 4. PROPIETATS DELS LOGARITMES  I) l II) l  log del producte = suma de log log del quocient = diferncia de log log2 ( 4 x 16 ) = log2 4 + log2 16 log10 (1000/100) = log10 1000 - log10 100 log2 4 = 2 ( (22 = 4) log10 1000 = 3 ( (103 = 1000) log2 16 = 4 ( (24 = 16) log10 100 = 2 ( (102 = 100) ----------------------------------- ---------------------------------------------- 6 = 4 + 2 1 = 3 2 log2 64 = 6 ( (26 = 64) log10 10 = 1 ( (101 = 10)   IV)  log de la potncia = potncia x log log de larrel = log / per lndex en realitat, log a (M n) = log a (M x M x M..) = en realitat, log a (M 1/n) = (1/n) x log M = log a (M) + log a (M) + log a (M)... = log2 V 64 = (1/2) x log2 64 = n x log a (M) log2 64 = 6 ( (26 = 64) : (2) log3 (81 2) = 2 x log3 81 ---------------------------------------------- log3 81 = 4 ( (34 = 81) 6 : 2 = 3 (x) 2 ----------------------------------- log2 8 = 3 ( (23 = 8) 8 = 4 x 2 log3 6561 = 8 ( (38 = 6561) Exercicis: Desenvolupar en forma logartmica A = a m x b n ( B = a x b p x c q ( C = a m / b n ( D = ( (a 2 x b 3 x c ) / (d 5 x e ) ) 4 ( E = V ( (a 2 x V b ) / ( V c 2 x d ) ) ( Exercicis: Desenvolupar els logaritmes 1. M = x 3 y 5 z 6 ( 2. N = x 2 z / y 3 ( 3. P = V ( x V y / z ) ( 4. Q = V x V y 4 V z ( 5. R = ( (V x3 V y ) / ( V z7 ) ) 4 6. S = V ( ( z V x 5 ) / (y + x ) 2 ) 5 Obtenir les expressions, A, B , C, D, E, i F Log A = 3 log x + 5 log y Log B = 7 log x (1/2) log y + log z Log C = log (a+b) + log (a-b) Log D = 3/2 log a + 5/3 log b 3 log c log d Log E = log a + (log b ) / m - log c (log d) / n Log F = 3/2 log x - log y - 5/14 log z 4.1 ALGUNES APLICACIONS DELS LOGARITMES a) Clcul de productes: A = 3,26 x 0,0217 x 32.458 = Log A = log (3,26 x 0,0217 x 32.458) ( Log A = log 3,26 + log 0,0217 + log 32.458 Log A = (0,51.. + (-1,66..) + 4,51.. ) = 3,36... ( A = ant-log ( 3,36..) = 2.296 Resoldre per logaritmes: 1. 0,00024826 * 3,49 * 5.628,46 = 0,02008 * 7,26854 * 426.814 = b) Clcul de quocients: A = 326,58 / 26,502 = Log A = log (326,58 / 26,502) ( Log A = log 326,58 - log 26,502 Log A = (2,514.. 1, 423..) = 1,0907... ( A = ant-log ( 1,0907..) = 12,32 Resoldre per logaritmes: 1. 0,000078504 : 0,0003729 = 3.264,49 : 0,026897 = c) Clcul de potncies: A = 9,026 7 = Log A = log (9,026 7 ) ( Log A = 7 x log 9,026 Log A = 7 x 0,955..) = 6,6885.. ( A = ant-log ( 6,6885..) = 4.881.000 Resoldre per logaritmes: 1. 9,02874 7 = 3.128,497 3 = d) Clcul darrels (potncies fraccionaries): A = V 82.346,37 = . 5 Log A = log (V 82.346,37 ) ( Log A = (1/5) x log 82.346,37 Log A = (1/5) x 4,9156.. = 0,7022.. ( A = ant-log ( 0,7022..) = 5,04 Resoldre per logaritmes: 1. V 7865,67 = 2. V 3.567,8 = 3. 34,567 (3/11) = d) Clcul doperacions combinades: A = ( (32,7 * 0,006487) / (5,6528) ) 4 = Log A = log ( (32,7 * 0,006487) / (5,6528) ) 4 ( Log A = 4 x ( log 32,7 + log 0,006487 log 5,6528) = 4 x ( 1,51..+(-2,18..) + 0,75..) Log A = -5,72... ( A = ant-log ( -5,72..) = 0,00000198 5. CANVIS DE BASES Si log a M = m ( ( a m = M log a (a m )  a m = b n log a (a m ) = log a (b n) log b M = n ( ( b n = M log a (b n ) per la propietat de les potncies ( m log a a = n log a b ( m x1 = n x log a b  com que m = log a M log a M = = log b M x log a b  n = log b M l og a M log b M = ------------ log a b log 457 2,65.... Exemple: log 7 457 = ???? ( log 7 457 = ----------- = ----------- = 3,14.... log 7 0,84... log 45 1,65.... log 0,39 45 = ???? ( log 0,39 45 = ----------- = ----------- = -4,04... log 0,39 -0,40.. 6. QESTIONS IMPORTANTS  I Atenci en les equacions log2 x ( log x2 Log2 x = log x ( log x = ( log x ) 2 Log x2 = log (x2) = 2 log x  II Un nombre sempre el podem transformar en logaritme 2 = log (100) 3,14 = log ( ant-log 3,14) = log 1.380,3843 5 = ln (ant ln 5) = ln = 148,413 III Solament es poden eliminar els logaritmes duna equaci quant es t  log dun membre = log de laltre membre ( log (3x 5) +.. = log 2 + x 5.. log x log 100 = log 25 ( log (x / 100) = log 25 ( ( log (x / 100) = log 25 x /100 = 25 ( x = 2.500  7. QUACIONS LOGARTMIQUES ( log f(x) = K  Tipus I log x + log 36 = log 612 log (36 ( x) = log 612 ( log (36 ( x) = log 612 ( 36 x = 612 ( x = 612/36 = 17 x = 17  Tipus II log 7x = log 37 + 5 log x log 7 + log x = log 37+ 5 log x ( log 7 log 37 = 5 log x - log x 0,84.. 1,56.. = 4 log x ( log x = - 0,72 / 4 = -0,18 ( log x = -018.. x = ant-log (-0,18..) ( x = 0,66..  Tipus III (log x) 2 log (972 / x) = 0 log 2 x - ( log 972 - log x ) = 0 ( log 2 x - log 972 + log x = 0 canvi de variable log x = t ( t2 + t 2,98.. = 0 t1 = 1,299 t2 = -2,299 desfer el canvi log x = 1,299.. ( x1 = ant-log (1,299..) = 19,92 log x = -2,299 ( x2 = ant-log (.2,299..) = 0,005 Exercicis: 1. 3 + log x = log 57 log 19 2 log x = log 25 3. log x 2 - log x log 17 = 0 4. log V 2x 3 + log (x + 2) = 1 + log 2 5. (x 2 x + 2 ) log 2 + log 250 = 3 6. log (26 - x 2) 2 log (4 x) = 0 7. log 2 x = log 9 2 8. log 7x = log 24 3 log x 9. (log x ) 2 1,7 = 0 10. log 2 log x log 5 = 0 11. log ( 17/x) + (log x ) 2 = 0 12. (log 9x log 37) / log 3 = (1 3 log x ) / log 2 7.1. SISTEMES DEQUACIONS LOGARTMIQUES  Tipus I x y = 15 log x + log y = 2,7  x y = 15 x-y = 15 x = 15 + y y2 +15 y 544 = 0 y1 = 17 log (x y ) = log (ant-log 2,7) x y = 544 (15+y) y = 544 y2 = -32  y2 = -32 no es soluci perqu log (-32) = ( ( y= 17 x = 15 + 17 = 32 x = 32  Tipus II 2 log x + 3 log y = 1,2 log x - log y = 0,8  2 log x + 3 log y = 1,2 2 log x + 3 log y = 1,2 log x = (3,9/5) log x - log y = 0,8 (3) 3 log x 3 log y = 2,6 log x = 0,78 ( x = ant-log 0,78 5 log x / = 3,9 x = 6,05 log 6,05 log y = 0,8 ( 0,78 log y = 0,8 ( log y = 0.78 0,8 ( log y = 0,02 y = ant-log 0,02 = 0,95 y = 0,95 Exercicis: 1. log x log y = 7 log x + log y = 3 2. x y = 19 log x - log y = 0,92 3. x2 - y2 = 19 log x + log y = 1 4. Demostrar la relaci entre a i b si 3 log a 5 log b = 0 5. La suma de dos nombres es 24 i la suma dels seus log en base 2 es 7. Calcular els nombres 8. EQUACIONS EXPONENCIALS ( a f (x) = K  Lequaci exponencial es la inversa de lequaci logartmica log a N = x (( a x = N  Tipus I 17 x = 36,5 Log 17 x = log 36,5 ( x log 17 = log 36,5 ( x = log 36,5 / log 17 = 1,2 x = 1,2  Tipus II log 19 7647 = x 19 x = 7647 ( log 19 x = log 7647 ( x log 19 = log 7647 ( x = log 7647/log 19 x= 3,03  x Tipus III 5 (x -1) = 36,5 2 2 5 (x -1) x = 15.625 ( 5 (x - x) = 15.625 ( 5 (x - x) = 5 6 ( x 2 - x = 6 ( x 2 - x 6 = 0 ( x1 = 3 x2 = -2  3 Tipus IV 16 (2x +1) = V 8 3 2 4 (2x +1) = V 2 3 ( 2 4 (2x +1) = 2 1 ( 8x + 4 = 1 ( 8x = - 3 ( x = - 3/8 Exercicis: 1. 6 x = 216 2. 2 3x-1 = 2.048 3. 3 2(x-1) = 81 5 4. V 2 3x - 3 = 64 5 5. 16 2x-1 = V 8 6. 6 x2 2x +1 = 1 7. 37 x 1 = 629,36 8. (17 3x-1 ) x = 22 x-3 9. 2 x 17 ( 2 x+1 = 288 = 0 10 3 2(x+1) = 18 ( 3 x - 9 x 11. V 6789,42 = 87,28 12. 3 x+3 + 3 x+2 + 3 x + 3 x-1 = 363 8.1. SISTEMES DEQUACIONS EXPONENCIALS  Tipus I 5 (x +y) = 36,2 6x = 7y  5 (x +y) = 36,2 log 5 (x +y) = log 36,2 (x+y) log 5 = log 36,5 x+y = log 36,5 / log 5 6x = 7y log 6 x = log 7 y x log 6 = y log 7 x log 6 = y log 7  x+y = 2,1.. x + y = 2,1 .. x = 1,12.. 0,7..x = 0,8..y 0,7x 0,8 y = 0 y = 1,03  Tipus II 2 x - 4 2y = 0 x y = 15  2 x = 2 2(2y) x = 4y 4y y = 15 3y = 15 y = 5 x = 4 * 5 x = 20  x y = 15 x y = 15 Exercicis: 1. 2 x 4 2y = 0 x y = 15 2. 2 x 3 y+1 = 113 x y = 0,5 3. 3 ( 10 x-1 5 y = 275 2 ( 10 x 3 ( 5 y+2 = 125 9. FUNCI LOGARITMICA EXPONENCIAL  9.1 Funci logartmica y = log a f(x) I Domini: dom (f) = x ( R+ = (0, () II Tall amb els eixos: eix X: sempre en x =1 eix Y: no talla mai III Simetries: no hi han mai IV Asmptotes: vertical en el propi eix Y  Y   a = 2 si y = 0 ( log 2 x = 0 ( 2 0 = x x = 1  y = 1 log 2 x = 1 ( 2 1 = x x= 2 y = 3 log 2 x = 3 ( 2 3 = x x= 8  X  y = -1 log 2 x = -1 ( 2 -1 = x x= 1/2  y = -5 log 2 x = -5 ( 2 -5 = x x= 1/32 a = 5 si y = 0 ( log 5 x = 0 ( 5 0 = x x = 1 y = 1 log 5 x = 1 ( 5 1 = x x= 5 a = 10 si y = 0 ( log 10 x = 0 ( 10 0 = x x = 1 y = 1 log 10 x = 1 ( 10 1 = x x= 10  Y y = log a x ( a ( (1, () ( a > 1   X  1 Y  a = 1/2 si y = 0 ( log 1/2 x = 0 ( 0 = x x = 1 y = 1 log 1/2 x = 1 ( 1 = x x= 1/2  X y = 3 log 1/2 x = 3 ( 3 = x x= 1/8   y = -1 log 1/2 x = -1 ( -1 = x x= 2  y = -5 log 1/2 x = -5 ( -5 = x x= 32  a = 1/5 y = 0 ( log 1/5 x = 0 ( 0 = x x = 1 y = 1 log 1/5 x = 1 ( 1/5 1 = x x= 1/5  y = -1 log 1/5 x = -1 ( 1/5 -1 = x x= 5  Y  1 X  y = log a x ( a ( (0, 1) ( a < 1 9.2 Funci exponencial y = a f(x) I Domini: dom (f) = x ( R = (-( , () II Tall amb els eixos: eix Y: sempre en y =1 eix X: no talla mai III Simetries: no hi han mai IV Asmptotes: horitzontal en el propi eix X  Y  a = 2 si x = 0 ( y = 2 0 ( y = 1  x = 1 y = 2 1 ( y = 2  x = 3 y = 23 ( y = 8  x = -1 y = 2-1 ( y = 1/2  x = -5 y = 2-5 ( y = 1/32  X a = 5 si x = 0 ( y = 5 0 ( y = 1 x = 1 y = 5 1 ( y = 5 a = 10 si x = 0 ( y =10 0 ( y = 1 x = 1 y = 10 1 ( y = 10  Y y =a (x) ( a ( (1, () ( a > 1   1 X    Y a = 1/2 si x = 0 ( y = 0 ( y = 1  x = 1 y = 1 ( y = 1/2  x = 3 y = 3 ( y = 1/8 x = -1 y = (1/2) 1 ( y = 2 x = -5 y = (1/2) 5 ( y = 32  X a = 1/5 si x= 0 ( y = (1/5) 0 ( y = 1  x = 1 y = (1/5) 1 ( y= 1/5   Y y = a x ( a ( (0, 1) ( a < 1  1 X  Funci logartmica funci exponencial Funci logartmica y = log a f(x) (( funci exponencial y = a f(x)   a > 1 a < 1   y = 2 x y = (1/2) x Y  Y y = log 2 x  X X   y = log 1/2 x Exercicis: Estudiar i representar les segents funscions 1. y = 5 x 7. y = log 4 x 2. y = 2 2x 8. y = log (1/3) 2x 3. y = (2/5) x 9. y = log 5x 4. y = 3 x 2 2x 10. y = log 2 3x 5. y = 3 x 2 -x 11. y = log V x 6. y = (1/5) -3x Determinar els valors de la variable dependent segons els valors de x 1. y = log 7 x x=1 3. y = log 8 x x = 512 2. y = log 5 x X 1/ 3.125 4. y = log 8 x x = 256 Determinar les funcions inverses de 1. y = 3 log 4 x 3. y = (1/5) log 5 x 2. y = 3 2x 4. y = V 7 5x Determinar si les funcions que es proposen son parells o senars 1. y = 2 x + (1/2) x 2. y = 2 - (1/2) x Problemes 1. Un bosc te una quantitat de fusta equivalent a 65 dam3. Aquest volum augmenta a ra dun 10% anual. Determinar la funci que representa aquesta situaci en el temps (y: volum de fusta, x: temps) 2. En el bosc anterior quina seria la quantitat de fusta disponible als 10 anys? Una mostra radioactiva pesa actualment 2 gr. .Per la emissi de radioactivitat es coneix que perd el 3% diari de radioactivitat. Determinar la funci que representa aquesta situaci en el temps (y: pes, x: temps) Un kg de la mostra radioactiva anterior, quant temps trigaria en reduir-se a la meitat? Un cultiu de virus de la grip, se sap que augmenta el 20 % cada dia. Al cap dun any i partint duna massa dun gram, quina massa sobtindr daquest microbi? 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