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By: Lulu Huang, Alison Li,Gladi Pang Period 4Logarithms By: Lulu Huang, Alison Li,Gladi Pang Period 4
8-4 Properties of LogarithmsProduct 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 LogarithmsExample 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 LogarithmsExample 1: log5XY = log5X + log 5Y Example 2: log3m4n-2 = log3 + logm4 +logn-2 = log3 + 4logm + -2logn
8-5 Solving Exponential EquationExample 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 FormulaChange 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 BaseExample 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 Equations8-5 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 EquationExample 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 LogarithmsExample 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 EquationsExample 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 EquationsExample 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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