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LOGARITHMS
                     exponential         logarithmic

                      m
               b             A      logb ( A)          m

                                   b>0
                                   A>0
Jeff Bivin -- LZHS
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)      y
Jeff Bivin -- LZHS
Evaluate
                     log5 (25)          u
                           u
                       5        25
                            u       2
                        5       5
                           u    2
                     log5 (25)          2

Jeff Bivin -- LZHS
Evaluate
                     log3 (81)         u
                         u
                       3       81
                           u       4
                       3       3
                        u      4
                     log3 (81)         4

Jeff Bivin -- LZHS
Evaluate                      1
                      log2      32        u
                            u        1
                        2            32
                            u        1
                        2            25

                        2u           2    5


                            u             5
                              1
                     log2    32               5
Jeff Bivin -- LZHS
Try this!!
                     log7 (7)           u
                           u
                       7        7
                           u        1
                       7        7
                        u       1
                     log7 (7)           1

Jeff Bivin -- LZHS
Try this!!
  Solve for x
                     logx(32) = 5
                      x5 = 32
                      x5 = 25
                       x = 2



Jeff Bivin -- LZHS
Try this!!
                        loga (1)
                            ? 0
                        a          1


                     loga (1)          0



Jeff Bivin -- LZHS
Try this!!
                        loga (a)
                            ? 1
                        a          a


                     loga (a)          1



Jeff Bivin -- LZHS
Special Logarithms

  log10 a   log a

        loge a      ln a
Properties of Logarithms
     Product Property
     Quotient Property
     Power Property




Jeff Bivin -- LZHS
Product Property
                              m       n       m n
                          a       a       a
                       multiplication     addition



  logb (m n)                      logb (m)            logb (n)
                multiplication                  addition


Jeff Bivin -- LZHS
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                6


Jeff Bivin -- LZHS
Quotient Property
                 m
               a     m                 n
      division
                 n
                   a                         subtraction
               a
                       m
        log ( )      b n        logb (m)       logb (n)
                     division              subtraction


Jeff Bivin -- LZHS
Quotient Property
                          32
                     log2 4      log2 (32)       log2 (4)
                                         5               2
                      log2 8     log2 (2 )       log2 (2 )
                            3
                     log2 (2 )       5               2
                           3                 3


Jeff Bivin -- LZHS
Power Property
                            m n       mn
                        a         a

                        p logb(m p )

                     logb(mp ) = p•logb(m)

Jeff Bivin -- LZHS
Power Property
                          7
                     log2 2       7 log2 (2)
                              7     71
                              7     7



Jeff Bivin -- LZHS
Power Property
                               n
                     log a a          n
                           n.log a ( a )
                          n.1
                          n

Jeff Bivin -- LZHS
Change of Base Formula
                               logb x
                     loga x
                               logb a


                                log x
                     log a x
                                log a
Jeff Bivin -- LZHS
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    m

Jeff Bivin -- LZHS
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
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
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      z2




Jeff Bivin -- LZHS
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 4


Jeff Bivin -- LZHS
Logarithm Equations
      Properties



            log a B   log a C   B C



Jeff Bivin -- LZHS
Solve for x
                     log3 3x    9       log3 x   3
                           3x       9   x    3
                                2x      12
                                    x   6



Jeff Bivin -- LZHS
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
Solve for x
                     log3 3x    9       log3 x   3
                           3x       9   x    3
                                2x      12
                                    x   6

                      6
Jeff Bivin -- LZHS
Solve for x
                     log4 7     log4 n   2     log4 6n
                              log4 7(n   2)    log4 6n
                                    7n    14   6n
                                          n    14




Jeff Bivin -- LZHS
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
Solve for x
                     log4 7     log4 n   2     log4 6n
                              log4 7(n   2)    log4 6n
                                    7n    14   6n
                                          n    14

                        14
Jeff Bivin -- LZHS
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           3

Jeff Bivin -- LZHS
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 positive

Jeff Bivin -- LZHS
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
                                                      3
Jeff Bivin -- LZHS
x            x   3
3       27   3       3          x   3
    x            x       2
5       49   5       7       ????????
Solve exponential equation with
logarithms
      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
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
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                   5
Jeff Bivin -- LZHS
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
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        x

Jeff Bivin -- LZHS

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Logarithma

  • 1.
  • 2. LOGARITHMS exponential logarithmic m b A logb ( A) m b>0 A>0 Jeff Bivin -- LZHS
  • 3. 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) y Jeff Bivin -- LZHS
  • 4. Evaluate log5 (25) u u 5 25 u 2 5 5 u 2 log5 (25) 2 Jeff Bivin -- LZHS
  • 5. Evaluate log3 (81) u u 3 81 u 4 3 3 u 4 log3 (81) 4 Jeff Bivin -- LZHS
  • 6. Evaluate 1 log2 32 u u 1 2 32 u 1 2 25 2u 2 5 u 5 1 log2 32 5 Jeff Bivin -- LZHS
  • 7. Try this!! log7 (7) u u 7 7 u 1 7 7 u 1 log7 (7) 1 Jeff Bivin -- LZHS
  • 8. Try this!! Solve for x logx(32) = 5 x5 = 32 x5 = 25 x = 2 Jeff Bivin -- LZHS
  • 9. Try this!! loga (1) ? 0 a 1 loga (1) 0 Jeff Bivin -- LZHS
  • 10. Try this!! loga (a) ? 1 a a loga (a) 1 Jeff Bivin -- LZHS
  • 11. Special Logarithms log10 a log a loge a ln a
  • 12. Properties of Logarithms  Product Property  Quotient Property  Power Property Jeff Bivin -- LZHS
  • 13. Product Property m n m n a a a multiplication addition logb (m n) logb (m) logb (n) multiplication addition Jeff Bivin -- LZHS
  • 14. 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 6 Jeff Bivin -- LZHS
  • 15. Quotient Property m a m n division n a subtraction a m log ( ) b n logb (m) logb (n) division subtraction Jeff Bivin -- LZHS
  • 16. Quotient Property 32 log2 4 log2 (32) log2 (4) 5 2 log2 8 log2 (2 ) log2 (2 ) 3 log2 (2 ) 5 2 3 3 Jeff Bivin -- LZHS
  • 17. Power Property m n mn a a p logb(m p ) logb(mp ) = p•logb(m) Jeff Bivin -- LZHS
  • 18. Power Property 7 log2 2 7 log2 (2) 7 71 7 7 Jeff Bivin -- LZHS
  • 19. Power Property n log a a n n.log a ( a ) n.1 n Jeff Bivin -- LZHS
  • 20. Change of Base Formula logb x loga x logb a log x log a x log a Jeff Bivin -- LZHS
  • 21. 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 m Jeff Bivin -- LZHS
  • 22. 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
  • 23. 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
  • 24. 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 z2 Jeff Bivin -- LZHS
  • 25. 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 4 Jeff Bivin -- LZHS
  • 26. Logarithm Equations Properties log a B log a C B C Jeff Bivin -- LZHS
  • 27. Solve for x log3 3x 9 log3 x 3 3x 9 x 3 2x 12 x 6 Jeff Bivin -- LZHS
  • 28. 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
  • 29. Solve for x log3 3x 9 log3 x 3 3x 9 x 3 2x 12 x 6 6 Jeff Bivin -- LZHS
  • 30. Solve for x log4 7 log4 n 2 log4 6n log4 7(n 2) log4 6n 7n 14 6n n 14 Jeff Bivin -- LZHS
  • 31. 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
  • 32. Solve for x log4 7 log4 n 2 log4 6n log4 7(n 2) log4 6n 7n 14 6n n 14 14 Jeff Bivin -- LZHS
  • 33. 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 3 Jeff Bivin -- LZHS
  • 34. 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 positive Jeff Bivin -- LZHS
  • 35. 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 3 Jeff Bivin -- LZHS
  • 36. x x 3 3 27 3 3 x 3 x x 2 5 49 5 7 ????????
  • 37. Solve exponential equation with logarithms 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
  • 38. 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
  • 39. 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 5 Jeff Bivin -- LZHS
  • 40. 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
  • 41. 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 x Jeff Bivin -- LZHS