Ejercicio 302

CAPITULO XXXVIII

Logaritmos
Ejercicio 302
Hallar el número de términos de las progresiones:
  1. ÷ ÷ 3 :6:………:48 Sea: u=48 a=3 r=2 n = loguloga logr +1 n = log48log3 log2 +1 n = 1.6812412370.4771212547 0.3010299957 +1 n =4+1 n =5
  2. ÷ ÷ 2 :3:………: 243 16 Sea: u= 243 16 a=2 r= 3 2 n= loguloga logr +1 n= log 243 16 log2 log 3 2 +1 n= log243log16log2 log3log2 +1 n= 2.3856062741.2041199830.3010299957 0.47712125471.204119983 +1 n=5+1 n=6
  3. ÷ ÷ 4 :8:………:512 Sea: u=512 a=4 r=2 n = loguloga logr +1 n = log512log4 log2 +1 n = 2.7092699610.6020599913 0.3010299957 +1 n =7+1 n =8
  4. ÷ ÷ 6 :8:………: 2048 81 Sea: u= 2048 81 a=6 r= 4 3 n= loguloga logr +1 n= log( 2048 81 ) log6 log( 4 3 ) +1 n= log2048log81log6 log4log3 +1 n= 3.3113299521.9084850190.7781512504 0.60205999130.4771212547 +1 n=5+1 n=6
  5. ÷ ÷ 2 :5:………: 625 8 Sea: u= 625 8 a=2 r= 5 2 n= loguloga logr +1 n= log( 625 8 ) log2 log( 5 2 ) +1 n= log625log8log2 log5log2 +1 n= 2.7958800170.9030899870.3010299957 0.69897000430.3010299957 +1 n=4+1 n=5

Ejercicio 301

CAPITULO XXXVIII

Logaritmos
Ejercicio 301
Resolver las ecuaciones:
  1. 5 x =3 log 5 x =log3 xlog5 =0.4771212547 x( 0.6989700043 ) =0.4771212547 x = 0.4771212547 0.6989700043 x =0.6826061945
  2. 7 x =512 log 7 x =log512 xlog7 =2.709269961 x( 0.84509805 ) =2.709269961 x = 2.709269961 0.84509805 x =3.205864684
  3. 0. 2 x =0.0016 log0. 2 x =log0.0016 xlog0.2 =2.795880017 x(0.6989700043 ) =2.795880017 x = 2.795880017 0.6989700043 x =4
  4. 9 x =0.576 log 9 x =log0.576 xlog9 =0.2395775166 x( 0.9542425094 ) =0.2395775166 x = 0.2395775166 0.9542425094 x =0.2510656507
  5. 3 x+1 =729 log 3 x+1 =log729 (x+1 ) log3 =2.862727528 (x+1 ) ( 0.4771212547 ) =2.862727528 x+1 = 2.862727528 0.4771212547 x =61 x =5
  6. 5 x2 =625 log 5 x2 =log625 (x2 ) log5 =2.795880017 (x2 ) ( 0.6989700043 ) =2.795880017 x2 = 2.795880017 0.6989700043 x =4+2 x =6
  7. 2 3x+1 =128 log 2 3x+1 =log128 (3x+1 ) log2 =2.10720997 (3x+1 ) ( 0.3010299957 ) =2.10720997 3x+1 = 2.10720997 0.3010299957 3x =71 x = 3 x =2
  8. 3 2x1 =2187 log 3 2x1 =log2187 (2x1 ) log3 =3.339848783 (2x1 ) ( 0.4771212547 ) =3.339848783 2x1 = 3.339848783 0.4771212547 2x =7+1 x = 2
  9. 1 1 2x =915 log1 1 2x =log915 (2x ) log11 =2.961421094 2x( 1.041392685 ) =2.961421094 2x = 2.961421094 1.041392685 2x =2.84371221 x = 2.84371221 2 x =1.421856105

Ejercicio 300

CAPITULO XXXVIII

Logaritmos
Ejercicio 300
Dados log 2 = 0.301030, log 3 = 0.477121, log 5 = 0.698970, log 7 = 0.845098
  1. log36 log36 =log( 2 2 × 3 2 ) =log 2 2 +log 3 2 =2log2+2log3 =2( 0.301030 ) +2( 0.477121 ) =0.60206+0.954242 =1.556302
  2. log75 log75 =log(3 × 5 2 ) =log3+log 5 2 =log3+2log5 =0.477121+2( 0.698970 ) =0.477121+1.39794 =1.875061
  3. log30 log30 =log(2 × 3 × 5 ) =log2+log3+log5 =0.301030+0.477121+0.698970 =1.477121
  4. log48 log48 =log( 2 4 × 3 ) =4log2+log3 =4( 0.301030 ) +0.477121 =1.20412+0.477121 =1.681241
  5. log120 log120 =log( 2 3 × 3 × 5 ) =3log2+log3+log5 =3( 0.301030 ) +0.477121+0.698970 =0.90309+1.176091 =2.079181
  6. log98 log98 =log( 7 2 × 2 ) =log 7 2 +log2 =2log7+0.301030 =2( 0.845098 ) +0.301030 =1.690196+0.301030 =1.991226
  7. log0.343 log0.343 =log( 343 1000 ) =log( 7 3 2 3 × 5 3 ) =log ( 7 ) 3 log( 2 3 × 5 3 ) =3log( 7 ) (log ( 2 ) 3 +log ( 5 ) 3 ) =3( 0.845098 ) (3log( 2 ) +3log( 5 ) ) =2.535294(3( 0.301030 ) +3( 0.698970 ) ) =0.464706
  8. log22.5 log22.5 =log( 45 2 ) =log( 3 2 × 5 2 ) =log( 3 2 × 5 ) log2 =log ( 3 ) 2 +log( 5 ) 0.301030 =2log( 3 ) +0.6989700.301030 =2( 0.477121 ) +0.39794 =1.352182
  9. log1.96 log1.96 =log( 49 25 ) =log( 7 2 5 2 ) =log ( 7 ) 2 log ( 5 ) 2 =2log( 7 ) 2log( 5 ) =2( 0.845098 ) 2( 0.698970 ) =1.6901961.39794 =0.292256
  10. log0.875 log0.875 =log( 7 8 ) =log( 7 2 3 ) =log7log 2 3 =0.8450983log2 =0.8450983( 0.301030 ) =0.8450980.90309 =0.057992
  11. log202.5 log202.5 =log( 405 2 ) =log( 3 4 × 5 2 ) =log( 3 4 × 5 ) log2 =log 3 4 +log50.301030 =4log3+0.6989700.301030 =4( 0.477121 ) +0.39794 =1.908484+0.39794 =2.306424
  12. log44.8 log44.8 =log( 224 5 ) =log( 2 5 × 7 5 ) =log( 2 5 × 7 ) log5 =log 2 5 +log70.698970 =5log2+0.8450980.698970 =5( 0.301030 ) +0.146128 =1.50515+0.146128 =1.651278
  13. log2 1 2 log2 1 2 =log( 5 2 ) =log5log2 =0.6989700.301030 =0.39794
  14. log1 1 2 log1 1 2 =log( 3 2 ) =log3log2 =0.4771210.301030 =0.176091
  15. log1 2 5 log1 2 5 =log( 7 5 ) =log7log5 =0.8450980.698970 =0.146128
  16. log2 1 3 log2 1 3 =log( 7 3 ) =log7log3 =0.8450980.477121 =0.367977
  17. Dado log 143 = 2.155336 y log 11 = 1.041393 hallar log 13.
    13 = 143 11 log13 =log( 143 11 ) =log143log11 =2.1553361.041393 =1.113943
  18. Dado log 225 = 2.352183 y log 9 = 0.954243 hallar log 25.
    25 = 225 9 log25 =log( 225 9 ) =log225log9 =2.3521830.954243 =1.39794

Ejercicio 299

CAPITULO XXXVIII

Logaritmos
Ejercicio 299
Hallar por log el valor de las expresiones siguientes:
  1. 515 × 78.19 6.13 515 × 78.19 6.13 =log( 515 × 78.19 6.13 ) =log(515 × 78.19 ) log6.13 =log515+log78.190.787460 =log515+log78.190.787460 =2.711807+1.8931510.787460 =3.817497968 =antilog( 3.817497968 ) =6568.980424
  2. 23.054 × 934.5 8164 23.054 × 934.5 8164 =log( 23.054 × 934.5 8164 ) =log(23.054 × 934.5 ) log( 8164 ) =log23.054+log934.53.911902996 =1.362746289+2.9705793063.911902996 =0.421422598 =antilog( 0.421422598 ) =2.638897967
  3. 8.14 × 9.73 0.6 × 7.8 8.14 × 9.73 0.6 × 7.8 =log( 8.14 × 9.73 0.6 × 7.8 ) =log(8.14 × 9.73 ) log(0.6 × 7.8 ) =log8.14+log9.73(log0.6+log7.8 ) =0.9106244049+0.9881128403(0.2218487496+0.8920946027 ) =1.228491392 =antilog( 1.228491392 ) =16.92354701
  4. 513.4 × 9.132 85.3 × 10.764 513.4 × 9.132 85.3 × 10.764 =log( 513.4 × 9.132 85.3 × 10.764 ) =log(513.4 × 9.132 ) log(85.3 × 10.764 ) =log513.4+log9.132(log85.3+log10.764 ) =2.710455864+0.9605659028(1.930949031+1.031973689 ) =0.7080990469 =antilog( 0.7080990469 ) =5.10621441
  5. 53.245 × 4325.6 32.815 × 91.79 53.245 × 4325.6 32.815 × 91.79 =log( 53.245 × 4325.6 32.815 × 91.79 ) =log(53.245 × 4325.6 ) log(32.815 × 91.79 ) =log53.245+log4325.6(log32.815+log91.79 ) =1.726278831+3.636046357(1.516072409+1.96279537 ) =1.883457409 =antilog( 1.883457409 ) =76.46406978
  6. 32.6 × (841.9 ) 0.017 × 732.14 Multiplico por 1 32.6 × 841.9 0.017 × 732.14 =log( 32.6 × 841.9 0.017 × 732.14 ) =log(32.6 × 841.9 ) log(0.017 × 732.14 ) =log32.6+log841.9(log0.017+log732.14 ) =1.5132176+2.92526051(1.769551079+2.864594135 ) =3.343435053 =antilog( 3.343435053 ) =2205.134344 32.6 × (841.9 ) 0.017 × 732.14 =2205.134344
  7. 95.36 × (0.14 ) (83.7 ) × ( 2.936 ) 95.36 × 0.14 83.7 × 2.936 =log( 95.36 × 0.14 83.7 × 2.936 ) =log(95.36 × 0.14 ) log(83.7 × 2.936 ) =log95.36+log0.14(log83.7+log2.936 ) =1.9793662420.8538719643(1.922725458+0.4677560512 ) =1.264987231 =antilog(1.264987231 ) =0.0543266304
  8. (7.2 ) × (8.135 ) (0.003 ) × ( 9134.7 ) = 7.2 × 8.135 0.003 × 9134.7 Multiplico por 1 7.2 × 8.135 0.003 × 9134.7 =log( 7.2 × 8.135 0.003 × 9134.7 ) =log(7.2 × 8.135 ) log(0.003 × 9134.7 ) =log7.2+log8.135(log0.003+log9134.7 ) =0.8573324964+0.9103575573(2.522878745+3.960694289 ) =0.3298745101 =antilog( 0.3298745101 ) =2.137344412 (7.2 ) × (8.135 ) (0.003 ) × ( 9134.7 ) =2.137344412
  9. 3 5 × 0. 2 4 3 5 × 0. 2 4 =log( 3 5 × 0. 2 4 ) =log 3 5 +log0. 2 4 =5log3+4log0.2 =5( 0.4771212547 ) +4(0.6989700043 ) =2.3856062742.795880017 =0.4102737437 =antilog(0.4102737437 ) =0.3888
  10. 5 1 2 × 3 2 3 5 1 2 × 3 2 3 =log( 5 1 2 × 3 2 3 ) =log 5 1 2 +log 3 2 3 = 1 2 log5+ 2 3 log3 = 1 2 ( 0.6989700043 ) + 2 3 ( 0.4771212547 ) =0.3494850022+0.3180808365 =0.6675658386 =antilog( 0.6675658386 ) =4.651208827
  11. 2 1 5 × 3 1 2 × 5 3 4 2 1 5 × 3 1 2 × 5 3 4 =log( 2 1 5 × 3 1 2 × 5 3 4 ) =log 2 1 5 +log 3 1 2 +log 5 3 4 = 1 5 log2+ 1 2 log3+ 3 4 log5 = 1 5 ( 0.3010299957 ) + 1 2 ( 0.4771212547 ) + 3 4 ( 0.6989700043 ) =0.06020599913+0.2385606274+0.5242275033 =0.8229941297 =antilog( 0.8229941297 ) =6.652641639
  12. 3 8 5. 6 5 3 8 5. 6 5 =log( 3 8 5. 6 5 ) =log ( 3 ) 8 log ( 5.6 ) 5 =8log35log5.6 =3.8169700383.740940135 =0.07602990273 =antilog( 0.07602990273 ) =1.191324032
  13. 0.5 3 7 2. 5 3 0.5 3 7 2. 5 3 =log( 0.5 3 7 2. 5 3 ) =log ( 0.53 ) 7 log ( 2.5 ) 3 =7log( 0.53 ) 3log( 2.5 ) =1.9300689131.193820026 =3.123888939 =antilog(3.123888939 ) =7.518151 × 1 0 4
  14. 3 2 5 2 5 3 3 2 5 2 5 3 =log( 3 2 5 2 5 3 ) =log 3 2 5 log 2 5 3 = 2 5 log3 5 3 log2 =0.19084850190.5017166594 =0.3108681576 =antilog(0.3108681576 ) =0.4888007262
  15. 7.86 × 8.14 (7.86 × 8.14 ) 1 2 =log (7.86 × 8.14 ) 1 2 = 1 2 log(7.86 × 8.14 ) = 1 2 (log7.86+log8.14 ) = 1 2 (0.895422546+0.9106244049 ) =0.9030234755 =antilog( 0.9030234755 ) =7.998774906
  16. 932.5 × 813.6 × 0.005 (932.5 × 813.6 × 0.005 ) 1 2 =log (932.5 × 813.6 × 0.005 ) 1 2 = 1 2 log(932.5 × 813.6 × 0.005 ) = 1 2 (log932.5+log813.6+log0.005 ) = 1 2 (2.96964884+2.910410942.301029996 ) =1.789514892 =antilog( 1.789514892 ) =61.59066488
  17. 93.7 × 104.2 8.35 × 7.3 ( 93.7 × 104.2 8.35 × 7.3 ) 1 2 =log ( 93.7 × 104.2 8.35 × 7.3 ) 1 2 = 1 2 log( 93.7 × 104.2 8.35 × 7.3 ) = 1 2 [log(93.7 × 104.2 ) log(8.35 × 7.3 ) ] = 1 2 [log93.7+log104.2(log8.35+log7.3 ) ] = 1 2 [1.971739591+2.017867719(0.9216864755+0.8633228601 ) ] =1.102298987 =antilog( 1.102298987 ) =12.65607347
  18. 23.725 × (9.182 ) × 7.184 3 = 23.725 × ( 9.182 ) × 7.184 3 Multiplico por 1 (23.725 × 9.182 × 7.184 ) 1 3 =log (23.725 × 9.182 × 7.184 ) 1 3 = 1 3 log(23.725 × 9.182 × 7.184 ) = 1 3 [log23.725+log9.182+log7.184 ] = 1 3 [1.375206221+0.9629372884+0.8563663237 ] =1.064836611 =antilog( 1.064836611 ) =11.61011739 23.725 × (9.182 ) × 7.184 3 =11.61011739
  19. 12316 × 0.25 931.8 × 0.07 4 ( 12316 × 0.25 931.8 × 0.07 ) 1 4 =log ( 12316 × 0.25 931.8 × 0.07 ) 1 4 = 1 4 log( 12316 × 0.25 931.8 × 0.07 ) = 1 4 [log(12316 × 0.25 ) log(931.8 × 0.07 ) ] = 1 4 [log12316+log0.25(log931.8+log0.07 ) ] = 1 4 [4.090469680.6020599913(2.9693227061.15490196 ) ] =0.4184972357 =antilog( 0.4184972357 ) =2.621182355
  20. 56813 22117 5 ( 56813 22117 ) 1 5 =log ( 56813 22117 ) 1 5 = 1 5 log( 56813 22117 ) = 1 5 [log( 56813 ) log( 22117 ) ] = 1 5 [4.7544477234.344726218 ] =0.08194430096 =antilog( 0.08194430096 ) =1.207658941
  21. ( 0.0316 0.1615 ) 3 2 ( 0.0316 0.1615 ) 3 2 =log ( 0.0316 0.1615 ) 3 2 = 3 2 log( 0.0316 0.1615 ) = 3 2 [log( 0.0316 ) log( 0.1615 ) ] = 3 2 [1.500312917+0.7918274733 ] =1.062728166 =antilog(1.062728166 ) =0.08655094896
  22. 3 3 4 5 2 3 3 3 4 5 2 3 =log( 3 3 4 5 2 3 ) =log ( 3 ) 3 4 log ( 5 ) 2 3 = 3 4 log( 3 ) 2 3 log( 5 ) = 3 4 ( 0.4771212547 ) 2 3 ( 0.6989700043 ) =0.1081390619 =antilog(0.1081390619 ) =0.7795804475
  23. 15 4 7 ( 15 4 ) 1 7 =log ( 15 4 ) 1 7 = 1 7 log( 15 4 ) = 1 7 [log( 15 ) log( 4 ) ] = 1 7 [1.1760912590.6020599913 ] =0.08200446682 =antilog( 0.08200446682 ) =1.207826258
  24. 5 3 5 = 5 3 5 Multiplico por 1 ( 5 3 ) 1 5 =log ( 5 3 ) 1 5 = 1 5 log( 5 3 ) = 1 5 [log( 5 ) log( 3 ) ] = 1 5 [0.69897000430.4771212547 ] =0.04436974992 =antilog( 0.04436974992 ) =1.107566343 5 3 5 =1.107566343
  25. ( 5 8 ) 6 5 ( 5 8 ) 6 5 =log ( 5 8 ) 6 5 = 6 5 log( 5 8 ) = 6 5 [log( 5 ) log( 8 ) ] = 6 5 [0.69897000430.903089987 ] =0.2449439792 =antilog(0.2449439792 ) =0.5689263134
  26. 3 5 × 5 7 3 ( 3 5 ) 1 2 × ( 5 7 ) 1 3 =log[ ( 3 5 ) 1 2 × ( 5 7 ) 1 3 ] =log ( 3 5 ) 1 2 +log ( 5 7 ) 1 3 = 1 2 log( 3 5 ) + 1 3 log( 5 7 ) = 1 2 [log( 3 ) log( 5 ) ] + 1 3 [log( 5 ) log( 7 ) ] = 1 2 [0.47712125470.6989700043 ] + 1 3 [0.69897000430.84509804 ] =0.15963372 =antilog(0.15963372 ) =0.6924147009
  27. 27 × 35 × 0.23 2 1 7 × 3 1 5 × ( 0.2 ) 1 3 =log[ 2 1 7 × 3 1 5 × ( 0.2 ) 1 3 ] =log ( 2 ) 1 7 +log ( 3 ) 1 5 +log ( 0.2 ) 1 3 = 1 7 log( 2 ) + 1 5 log( 3 ) + 1 3 log( 0.2 ) = 1 7 ( 0.3010299957 ) + 1 5 ( 0.4771212547 ) + 1 3 (0.6989700043 ) =0.09456146541 =antilog(0.09456146541 ) =0.8043379031
  28. 32.14 × 59.33 317.64 ( 32.14 ) 1 2 × ( 59.3 ) 1 3 ( 317.6 ) 1 4 =log[ ( 32.14 ) 1 2 × ( 59.3 ) 1 3 ( 317.6 ) 1 4 ] =log( ( 32.14 ) 1 2 × ( 59.3 ) 1 3 ) log ( 317.6 ) 1 4 =log ( 32.14 ) 1 2 +log ( 59.3 ) 1 3 1 4 log( 317.6 ) = 1 2 log( 32.14 ) + 1 3 log( 59.3 ) 1 4 ( 2.501880494 ) = 1 2 ( 1.507045872 ) + 1 3 ( 1.773054693 ) 1 4 ( 2.501880494 ) =0.7190710439 =antilog( 0.7190710439 ) =5.236860966
  29. ( 0.75 ) 2 × 39.162 0.07 × 3.89 ( ( 0.75 ) 2 × 39.162 0.07 × 3.89 ) 1 2 =log ( ( 0.75 ) 2 × 39.162 0.07 × 3.89 ) 1 2 = 1 2 log( ( 0.75 ) 2 × 39.162 0.07 × 3.89 ) = 1 2 [log( ( 0.75 ) 2 × 39.162 ) log(0.07 × 3.89 ) ] = 1 2 [log ( 0.75 ) 2 +log( 39.162 ) (log( 0.07 ) +log( 3.89 ) ) ] = 1 2 [2(0.1249387366 ) +1.592864863(1.15490196+0.5899496013 ) ] =0.9539698743 =antilog( 0.9539698743 ) =8.994351882
  30. ( 0.2 ) 3 × ( 0.3 ) 2 ( 0.05 ) 4 × 3.25 ( ( 0.2 ) 3 × ( 0.3 ) 2 ( 0.05 ) 4 × 3.25 ) 1 2 =log ( ( 0.2 ) 3 × ( 0.3 ) 2 ( 0.05 ) 4 × 3.25 ) 1 2 = 1 2 log( ( 0.2 ) 3 × ( 0.3 ) 2 ( 0.05 ) 4 × 3.25 ) = 1 2 [log( ( 0.2 ) 3 × ( 0.3 ) 2 ) log( ( 0.05 ) 4 × 3.25 ) ] = 1 2 [log ( 0.2 ) 3 +log ( 0.3 ) 2 (log ( 0.05 ) 4 +log( 3.25 ) ) ] = 1 2 [3log( 0.2 ) +2log( 0.3 ) (4log( 0.05 ) +log( 3.25 ) ) ] = 1 2 [3(0.6989700043 ) +2(0.5228787453 ) (4(1.301029996 ) +0.511883361 ) ] =0.7747845591 =antilog( 0.7747845591 ) =5.95366726

Ejercicio 298

CAPITULO XXXVIII

Logaritmos
Ejercicio 298
Hallar el valor de las siguientes expresiones por medio de logaritmos:
  1. 532 × 0.184 532 × 0.184 =log(532 × 0.184 ) =log532+log0.184 =2.7259116320.735182177 =1.99072955 =antilog1.99072955 =97.888
  2. 191.7 × 432 191.7 × 432 =log(191.7 × 432 ) =log191.7+log432 =2.282622113+2.635483747 =4.91810586 =antilog4.91810586 =82814.4
  3. 0.7 × 0.013 × 0.9 0.7 × 0.013 × 0.9 =log(0.7 × 0.013 × 0.9 ) =log0.7+log0.013+log0.9 =2.086716098 =antilog2.086716098 =8.19 × 1 0 3 =0.00819
  4. 7.5 × 8.16 × 0.35 × 10037 7.5 × 8.16 × 0.35 × 10037 =log(7.5 × 8.16 × 0.35 × 10037 ) =log7.5+log8.16+log0.35+log10037 =5.332423391 =antilog5.332423391 =214992.54
  5. 3.2 × 4.3 × 7.8 × 103.4 × 0.019 3.2 × 4.3 × 7.8 × 103.4 × 0.019 =log(3.2 × 4.3 × 7.8 × 103.4 × 0.019 ) =log3.2+log4.3+log7.8+log103.4+log0.019 =2.323987176 =antilog2.323987176 =210.8566
  6. 95.13 ÷ 7.23 95.13 ÷ 7.23 =log(95.13 ÷ 7.23 ) =log95.13log7.23 =1.119179199 =antilog1.119179199 =13.15767635
  7. 8.125 ÷ 0.9324 8.125 ÷ 0.9324 =log(8.125 ÷ 0.9324 ) =log8.125log0.9324 =0.9402211048 =antilog0.9402211048 =8.714071214
  8. 7653.95 ÷ 12.354 7653.95 ÷ 12.354 =log(7653.95 ÷ 12.354 ) =log7653.95log12.354 =2.792078024 =antilog2.792078024 =619.5523717
  9. 0.72183 0.0095 0.72183 0.0095 =log( 0.72183 0.0095 ) =log0.72183log0.0095 =1.880711323 =antilog1.880711323 =75.98210526
  10. 9114 0.02 9114 0.02 =log( 9114 0.02 ) =log9114log0.02 =5.658679029 =455700
  11. 2 10 2 10 =log 2 10 =10log2 =10( 0.3010299957 ) =3.010299957 =antilog3.010299957 =1024
  12. 0.1 5 3 0.1 5 3 =log0.1 5 3 =3log0.15 =3(0.8239087409 ) =2.471726223 =antilog(2.471726223 ) =3.375 × 1 0 3
  13. 18.6 5 4 18.6 5 4 =log18.6 5 4 =4log18.65 =4( 1.270678836 ) =5.082715345 =antilog( 5.082715345 ) =120980.4915
  14. 20.8 4 2 20.8 4 2 =log20.8 4 2 =2log20.84 =2( 1.318897715 ) =2.637795429 =antilog( 2.637795429 ) =434.3056
  15. 7. 2 6 7. 2 6 =log7. 2 6 =6log7.2 =6( 0.8573324964 ) =5.143994979 =antilog( 5.143994979 ) =139314.0695
  16. 3 3 =log 3 1 2 = 1 2 log3 = 1 2 ( 0.4771212547 ) =0.2385606274 =antilog( 0.2385606274 ) =1.732050808
  17. 23 23 =log 2 1 3 = 1 3 log( 2 ) = 1 3 ( 0.3010299957 ) =0.1003433319 =antilog( 0.1003433319 ) =1.25992105
  18. 54 54 =log 5 1 4 = 1 4 log5 = 1 4 ( 0.6989700043 ) =0.1747425011 =antilog( 0.1747425011 ) =1.495348781
  19. 635 635 =log6 3 1 5 = 1 5 log63 = 1 5 ( 1.799340549 ) =0.3598681099 =antilog( 0.3598681099 ) =2.290172049
  20. 8157 8157 =log81 5 1 7 = 1 7 log815 = 1 7 ( 2.911157609 ) =0.4158796584 =antilog( 0.4158796584 ) =2.605431493