# By: Lulu Huang, Alison Li,Gladi Pang Period 4

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By: Lulu Huang, Alison Li,Gladi Pang Period 4
Logarithms By: Lulu Huang, Alison Li,Gladi Pang Period 4

8-4 Properties of Logarithms
Product Property:    logbXY  =   logbX  +  logbY Quotient Property :   logbX   =  logbX  -  logbY                                     Y Power property:       logbXy  =   ylogbX

8-4 Identifying Properties
Example 1: log5  +   log6  =  log 30        product property Example 2: log55  +  log520  -  log 54  =   log525     product and quotient property

8-4 Simplifying Logarithms
Example 1: log44    +   log432 = log4 (4 x 32) = log4132 Example 2: log 7X   +   log7Y   -  log 7Z = log 7  (X x Y)           Z = log 7  XY         Z

8-4 Expanding Logarithms
Example 1: log5XY = log5X  +  log 5Y Example 2: log3m4n-2 = log3  + logm4 +logn-2 = log3 +  4logm  + -2logn

8-5 Solving Exponential Equation
Example 1: 72X  =  25 log72X = log25 2Xlog7  =   log25 log7            log 7 2X  =              2 X  =  0.8271 Example 2: 202X+1  =  260 log202X+1  =  log260 2X+1log20  = log260 log 20             log 20 2X+1  =       -1         -1 2X  =               2 X =  0.4281

8-5 Using Change Of Base Formula
Change Of Base Formula: logaN  = log N               log a Example 1: log333 = log33    log 3 Example 2: log5135 = log135    log5

8-5 Solving Exponential Equations by Changing Base
Example 1: 2X =  5 log22X = log25 Xlog22 = log25 Xlog2   =    log5 log2            log2 X =  2.322 Example 2: 73X+4 =  79 log773X+4   =   log779 3X + 4 log7  =     log79           log7          log7 3X + 4 =          -4        -4 3X  =            3 x =

8-5 Solving a Logarithmic Equations
8-5 Solving a Logarithmic Equations Example 1: log2X =  5 10log 2X =  105 2X = 105 2X = 10000 2         2 X =  50000 Example 2: 2log X = 2 log X2  =  2 10log X2 = 102 X2 = 102 X  = 1002 X  = 10000 2 X2 1002 =

8-5 Using Logarithmic Properties to Solve Equation
Example 1: log X - log 3 =  3 log X    = 3      3 10log X  = 103           3    3 x X      = 1000  (3)       3 X = 3000 Example 2: log2 X - log2 6 + log2 2 = 3 log 2 2X   = 3         6 2 log2  2X  =  23              6 6 x 2X  = 8 (6)      6 2X = 48 2      2 X =  24

8-6 Simplifying Natural Logarithms
Example 1: 3 ln 5 ln 53 = ln125 =  4.83 Example 2: ln a - 2 ln b  +  2 ln c = ln a - ln b2 + ln c2 = ln a x c2         b2 = ln ac2        b2

8-6 Solving Natural Logarithmic Equations
Example 1: ln 3X = 6 e ln 3X = e6 3X  =           3 X = Example 2: ln 2X =               - 1.1 ln 2X  =  10.9 e ln 2X = e10.9 2X =          2 X =

8-6 Solving Exponential Equations
Example 1: eX = 5 ln e X= ln 5 X =  1.609 Example 2: 2e2x  - 7 = 53          +7    +7 2e2X  = 60 2           2 e2X   =   30 ln e 2x  = ln 30 2X  =        2 X =

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