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Published byGrace Lorraine Andrews Modified over 9 years ago
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Logarithms and Their Properties Lesson 4.1
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Recall the Exponential Function General form Given the exponent what is the resulting y-value? Now we look at the inverse of this function Now we will ask, given the result, what exponent is needed to achieve it?
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A New Function Consider the exponential function y = 10 x Based on that function, declare a new function x = log 10 y You should be able to see that these are inverse functions In general The log of a number is an exponent
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The Log Function Try These log 3 9 = ?log 2 32 = ? log 0.01 = ? Note: if no base specified, default is base of 10
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Properties of Logarithms Note box on page 154 of text Most used properties
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Natural Logarithms We have used base of 10 for logs Another commonly used base for logs is e e is an irrational number (as is ) e has other interesting properties Later to be discovered in calculus Use ln button on your calculator
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Properties of the Natural Logarithm Recall that y = ln x x = e y Note that ln 1 = 0 and ln e = 1 ln (e x ) = x (for all x) e ln x = x (for x > 0) As with other based logarithms
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Use Properties for Solving Exponential Equations Given Take log of both sides Use exponent property Solve for what was the exponent Note this is not the same as log 1.04 – log 3
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Misconceptions log (a+b) NOT the same as log a + log b log (a-b) NOT the same as log a – log b log (a * b) NOT same as (log a)(log b) log (a/b) NOT same as (log a)/(log b) log (1/a) NOT same as 1/(log a)
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Assignment Lesson 4.1 Page 157 Exercises 1 – 51 odd
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