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LOGARITHMS                     exponential         logarithmic                      m               b             A      l...
exponential            logarithmic                             m                         b       A        logb ( A)    m  ...
Evaluate                     log5 (25)          u                           u                       5        25           ...
Evaluate                     log3 (81)         u                         u                       3       81               ...
Evaluate                      1                      log2      32        u                            u        1          ...
Try this!!                     log7 (7)           u                           u                       7        7          ...
Try this!!  Solve for x                     logx(32) = 5                      x5 = 32                      x5 = 25        ...
Try this!!                        loga (1)                            ? 0                        a          1             ...
Try this!!                        loga (a)                            ? 1                        a          a             ...
Special Logarithms  log10 a   log a        loge a      ln a
Properties of Logarithms     Product Property     Quotient Property     Power PropertyJeff Bivin -- LZHS
Product Property                              m       n       m n                          a       a       a              ...
Product Property           log2 (16 4)           log2 (16)       log2 (4)                        4   2            4       ...
Quotient Property                 m               a     m                 n      division                 n               ...
Quotient Property                          32                     log2 4      log2 (32)       log2 (4)                    ...
Power Property                            m n       mn                        a         a                        p logb(m ...
Power Property                          7                     log2 2       7 log2 (2)                              7     7...
Power Property                               n                     log a a          n                           n.log a ( ...
Change of Base Formula                               logb x                     loga x                               logb ...
log b       1         1        log a b                         log a    log a      logb a                                 ...
Expand                         3   2                           log5 ( x y )                                          3    ...
Expand                             x7                             log   5 y2 z5                                           ...
Condense                     5 log3 x   6 log3 y          2 log3 z        power property          log3 x   5              ...
Condense       1                     2   log10 x    2 log10 y           4 log10 z                                         ...
Logarithm Equations      Properties            log a B   log a C   B CJeff Bivin -- LZHS
Solve for x                     log3 3x    9       log3 x   3                           3x       9   x    3               ...
Solve for x                     log3 3x     9       log3 x      3                            check       x       6        ...
Solve for x                     log3 3x    9       log3 x   3                           3x       9   x    3               ...
Solve for x                     log4 7     log4 n   2     log4 6n                              log4 7(n   2)    log4 6n   ...
Solve for x                     log4 7       log4 n   2       log4 6n                              check        n     14  ...
Solve for x                     log4 7     log4 n   2     log4 6n                              log4 7(n   2)    log4 6n   ...
Solve for x                     log2 x 1              log2 x 1      3                                                     ...
Solve for x                     log2 x 1         log2 x 1             3         check x             3                 chec...
Solve for x                     log2 x 1       log2 x 1      3                          log2 ( x 1)(x 1)        3         ...
x            x   33       27   3       3          x   3    x            x       25       49   5       7       ????????
Solve exponential equation withlogarithms      x                   x       2  5        49         5       7        ???????...
Solve for x                              4x                 2                     log 5             log 7                 ...
Solve for x                             x 2              x 1                      log 3          log 5                    ...
Try this                                          3x 1                            5        7 2                            ...
Try this                                       3x 2                       ln 15      ln e                         ln(15)  ...
Logarithma
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Logarithma

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Logarithma

  1. 1. LOGARITHMS exponential logarithmic m b A logb ( A) m b>0 A>0Jeff Bivin -- LZHS
  2. 2. exponential logarithmic m b A logb ( A) m 2 3 9 log3 (9) 2 53 125 log5 (125) 3 3 1 1 2 8 log2 8 3 1 5 2 32 log1 (32) 5 2 y x 2 log2 ( x) yJeff Bivin -- LZHS
  3. 3. Evaluate log5 (25) u u 5 25 u 2 5 5 u 2 log5 (25) 2Jeff Bivin -- LZHS
  4. 4. Evaluate log3 (81) u u 3 81 u 4 3 3 u 4 log3 (81) 4Jeff Bivin -- LZHS
  5. 5. Evaluate 1 log2 32 u u 1 2 32 u 1 2 25 2u 2 5 u 5 1 log2 32 5Jeff Bivin -- LZHS
  6. 6. Try this!! log7 (7) u u 7 7 u 1 7 7 u 1 log7 (7) 1Jeff Bivin -- LZHS
  7. 7. Try this!! Solve for x logx(32) = 5 x5 = 32 x5 = 25 x = 2Jeff Bivin -- LZHS
  8. 8. Try this!! loga (1) ? 0 a 1 loga (1) 0Jeff Bivin -- LZHS
  9. 9. Try this!! loga (a) ? 1 a a loga (a) 1Jeff Bivin -- LZHS
  10. 10. Special Logarithms log10 a log a loge a ln a
  11. 11. Properties of Logarithms  Product Property  Quotient Property  Power PropertyJeff Bivin -- LZHS
  12. 12. Product Property m n m n a a a multiplication addition logb (m n) logb (m) logb (n) multiplication additionJeff Bivin -- LZHS
  13. 13. Product Property log2 (16 4) log2 (16) log2 (4) 4 2 4 2 log2 (2 2 ) log2 (2 ) log2 (2 ) 6 log2 (2 ) 4 2 6 6Jeff Bivin -- LZHS
  14. 14. Quotient Property m a m n division n a subtraction a m log ( ) b n logb (m) logb (n) division subtractionJeff Bivin -- LZHS
  15. 15. Quotient Property 32 log2 4 log2 (32) log2 (4) 5 2 log2 8 log2 (2 ) log2 (2 ) 3 log2 (2 ) 5 2 3 3Jeff Bivin -- LZHS
  16. 16. Power Property m n mn a a p logb(m p ) logb(mp ) = p•logb(m)Jeff Bivin -- LZHS
  17. 17. Power Property 7 log2 2 7 log2 (2) 7 71 7 7Jeff Bivin -- LZHS
  18. 18. Power Property n log a a n n.log a ( a ) n.1 nJeff Bivin -- LZHS
  19. 19. Change of Base Formula logb x loga x logb a log x log a x log aJeff Bivin -- LZHS
  20. 20. log b 1 1 log a b log a log a logb a log b log b log c log c log a b.logb c . log a c log a log b log a n n log b n.log b n log am b m .log a b log a m.log a mJeff Bivin -- LZHS
  21. 21. Expand 3 2 log5 ( x y ) 3 2 product property log5 ( x ) log5 ( y ) power property 3 log5 ( x) 2 log5 ( y)Jeff Bivin -- LZHS
  22. 22. Expand x7 log 5 y2 z5 7 2 5 quotient property log5 ( x ) log5 ( y z ) product property log5 ( x )7 2 log5 ( y ) 5 log5 ( z )distributive property log5 ( x )7 2 log5 ( y ) 5 log5 ( z ) power property 7 log5 ( x) 2 log5 ( y) 5 log5 ( z)Jeff Bivin -- LZHS
  23. 23. Condense 5 log3 x 6 log3 y 2 log3 z power property log3 x 5 log3 y 6 log3 z 2 product property log3 x y 5 6 log3 z 2 x5 y 6 quotient property log 3 z2Jeff Bivin -- LZHS
  24. 24. Condense 1 2 log10 x 2 log10 y 4 log10 z 1 2 4 Power property log10 x 2 log10 y log10 z 1 2 4 group / factor log10 x 2 log10 y log10 z 1 product property 2 4 log10 x 2 log10 y z 1 quotient property log10 x2 log x y2z4 10 y 2 z 4Jeff Bivin -- LZHS
  25. 25. Logarithm Equations Properties log a B log a C B CJeff Bivin -- LZHS
  26. 26. Solve for x log3 3x 9 log3 x 3 3x 9 x 3 2x 12 x 6Jeff Bivin -- LZHS
  27. 27. Solve for x log3 3x 9 log3 x 3 check x 6 log3 3(6) 9 log3 6 3 log3 18 9 log3 6 3 log3 9 log3 9 checks!Jeff Bivin -- LZHS
  28. 28. Solve for x log3 3x 9 log3 x 3 3x 9 x 3 2x 12 x 6 6Jeff Bivin -- LZHS
  29. 29. Solve for x log4 7 log4 n 2 log4 6n log4 7(n 2) log4 6n 7n 14 6n n 14Jeff Bivin -- LZHS
  30. 30. Solve for x log4 7 log4 n 2 log4 6n check n 14 log4 7 log4 14 2 log4 6(14) log4 7 log4 12 log4 84 log4 7(12) log4 84 log4 84 log4 84 checks!Jeff Bivin -- LZHS
  31. 31. Solve for x log4 7 log4 n 2 log4 6n log4 7(n 2) log4 6n 7n 14 6n n 14 14Jeff Bivin -- LZHS
  32. 32. Solve for x log2 x 1 log2 x 1 3 3 log 2 ( x 1)( x 1) log 2 2 3 ( x 1)( x 1) 2 2 x 1 8 2 x 9 x 3Jeff Bivin -- LZHS
  33. 33. Solve for x log2 x 1 log2 x 1 3 check x 3 check x 3 log2 3 1 log2 3 1 3 log2 3 1 log2 3 1 3 log2 4 log2 2 3 log2 2 log2 4 3 2 1 3 fails 3 3 The argument checks! must be positiveJeff Bivin -- LZHS
  34. 34. Solve for x log2 x 1 log2 x 1 3 log2 ( x 1)(x 1) 3 3 2 ( x 1)(x 1) 2 8 x 1 2 9 x 3 x 3Jeff Bivin -- LZHS
  35. 35. x x 33 27 3 3 x 3 x x 25 49 5 7 ????????
  36. 36. Solve exponential equation withlogarithms x x 2 5 49 5 7 ???????? log 5 x log 7 2 x.log 5 2.log 7 log 7 x 2. log 5 x 2.log 5 7
  37. 37. Solve for x 4x 2 log 5 log 7 (4 x) log(5) (2) log(7) 4 log(5) 4 log(5) 2 log( 7 ) x 4 log(5 ) log(7 2 ) x log(54 ) x log 625 (49)Jeff Bivin -- LZHS
  38. 38. Solve for x x 2 x 1 log 3 log 5 ( x 2) log(3) ( x 1) log(5) x log(3) 2 log(3) x log(5) 1log(5) x log(3) x log(5) log(5) 2 log(3) x log(3) log(5) log(5) 2 log(3) 5 log( ) 5 x 9 3 x log 3 ( ) log( ) ( ) 9 5 5Jeff Bivin -- LZHS
  39. 39. Try this 3x 1 5 7 2 5 3x 1 log 7 log 2 5 log(7 ) (3x 1) log(2) 5 log( 7 ) 3x log(2) 1 log(2) 5 log(7 ) log(2) 3x log(2) log( 10 ) log(8) 7 x 10 log ( )Jeff Bivin -- LZHS 8 7 x
  40. 40. Try this 3x 2 ln 15 ln e ln(15) 1 (3x 2) ln(e) ln(15) 3x 2 ln(15) 2 3x ln(15) 2 3 xJeff Bivin -- LZHS

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