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Logarithms

Logarithms. By: Lulu Huang, Alison Li,Gladi Pang Period 4. Product Property :    log b XY  =   log b X  +  log b Y Quotient Property :    log b X   =  log b X  -  log b Y                                     Y Power property :       log b X y  =    y logbX. 8-4 Properties of Logarithms.

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Logarithms

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

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

  3. 8-4  Identifying Properties   • Example 1:log5  +   log6  =  log 30        • product property • Example 2: log55  +  log520  -  log 54  =   log525     • product and quotient property

  4. 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 Simplifying Logarithms

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

  6. Example 1:72X  =  25 log72X = log25 2Xlog7  =   log25 log7            log 7 2X  =  1.65422            2 X  =  0.8271 Example 2:202X+1  =  260 log202X+1  =  log260 2X+1log20  = log260log 20             log 20 2X+1  =   1.8562   -1         -1 2X  =   0.85622            2 X =  0.4281 8-5 Solving Exponential Equation

  7. Change Of Base Formula:logaN  = log Nlog a Example 1: log333 = log33log 3Example 2:log5135 = log135log5 8-5 Using Change Of Base Formula

  8. Example 1: 2X =  5 log22X = log25 Xlog22 = log25 Xlog2   =    log5log2            log2 X =  2.322 Example 2:73X+4 =  79 log773X+4   =   log779 3X + 4 log7  =     log79log7          log7 3X + 4 =  2.2455    -4        -4 3X  =  -1.75453          3 x = -0.584 8-5 Solving Exponential Equations by Changing Base

  9. 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 8-5 Solving a Logarithmic Equations 2 X2 1002 =

  10. Example 1:log X - log 3 =  3 log X    = 3  3 10logX  = 103          3    3 x X      = 1000  (3)3 X = 3000 Example 2: log2 X - log2 6 + log2 2 = 3 log 22X   = 3        6 2 log22X  =  23             6 6 x 2X  = 8 (6)6 2X = 482      2 X =  24 8-5 Using Logarithmic Properties to Solve Equation

  11. 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 Simplifying Natural Logarithms

  12. Example 1: ln 3X = 6 e ln 3X = e6 3X  =  403.433         3 X = 134.48 Example 2:1.1 + ln 2X = 12-1.1               - 1.1 ln 2X  =  10.9 e ln 2X = e10.9 2X =  54176.3642        2 X =  27088.182 8-6 Solving Natural Logarithmic Equations

  13. Example 1:eX = 5 ln e X= ln 5 X =  1.609 Example 2:2e2x  - 7 = 53+7    +7 2e2X  = 602           2 e2X   =   30 ln e2x  = ln 30 2X  = 3.40122        2 X = 1.7006 8-6 Solving Exponential Equations

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