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Logarithms and Their PropertiesRecall the Exponential FunctionA New FunctionThe Log FunctionProperties of LogarithmsNatural LogarithmsProperties of the Natural LogarithmUse Properties for Solving Exponential EquationsMisconceptionsAssignmentLogarithms and Their PropertiesLesson 5.1Recall the Exponential Function•General formGiven the exponentwhat is the resulting y-value?•Now we look at the inverse of this functionNow we will ask, given the result, what exponent is needed to achieve it?( )xf x a b= �A New Function•Consider the exponential function y = 10x•Based on that function, declare a new functionx = log10y•You should be able to see that these are inverse functions•In general•The log of a numberis an exponentlogxba b a x= � =The Log Function•Try Theselog39 = ? log232 = ? log 0.01 = ?Note: if no base specified, default is base of 10233 9log 9 2so==522 32log 32 5so==210 0.01log 0.01 2so-==-Properties of Logarithms•Note box on page 154 of text•Most used propertieslog( ) log loglog log loglog logna b a baa bba n a� =� �=�=-�� ��+Natural Logarithms•We have used base of 10 for logs•Another commonly used base for logs is ee is an irrational number (as is ) •e has other interesting propertiesLater to be discovered in calculus•Use ln button on your calculatorpProperties of the Natural Logarithm•Recall that y = ln x x = ey•Note thatln 1 = 0 and ln e = 1ln (ex) = x (for all x)e ln x = x (for x > 0)•As with other based logarithmsln( ) ln lnln ln lnln lnna b a baa bba n a� = +� �= -� �� �= �Use Properties for Solving Exponential Equations•Given•Take log ofboth sides•Use exponent property•Solve for whatwas the exponent1.04 3t=( )log 1.04 log 3t=log(1.04) log 3t � =log 3log1.04t =Note this is not the same aslog 1.04 – log 3Misconceptions•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)Assignment•Lesson 5.1•Page 185•Exercises 1 – 51

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