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This 6 page Class Notes was uploaded by Aileen Davis on Tuesday October 20, 2015. The Class Notes belongs to MATH0000D at Sierra College taught by Frederick in Fall. Since its upload, it has received 22 views. For similar materials see /class/225367/math0000d-sierra-college in Mathematics (M) at Sierra College.
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Date Created: 10/20/15
Math D 93 As we have seen exponential functions pass the horizontal line test This means exponential functions have inverses Lets apply our inverse nding strategy to exponential functions f96 bm We dont have the tools to solve such a problem Thus we will develop a tool called the logarithm De nition 01 Fora gt 0 andb gt 0 b 31 1 y logbx is equivalent to by i The function fi logbx is the logarithmic function with base b The equations y logbx and by z are different ways of expressing the same information They are7 respectively7 the logarithmic and exponential forms Like with other equations we sometimes have advantages displaying the equations in various forms Speci cally7 in our example above we were unable to isolate the exponent thus we could not solve for it In the logarithmic form we can isolate the exponent y and thus solve for it Example 01 Write the logarithm in its equivalent emponentialform o 6 l09264 means 26 64 o 2 loggi means 92 z o 3 logb27 means b3 27 o y log5125 means 5y 125 Example 02 Wiite the eaponential in its equivalent logarithmic form 0 5 3 5 means log5 73 o m 4 means l09644 o 152 z means logl5x 2 o b3 343 means logb343 3 0 8y 300 means logb300 y Example 03 Evaluate some simple logarithms o log749 means 77 49 is 2 o loga means 67 is 71 o logy6 means 67 V5 is i o lognll means 117 11 is 1 o 0961 means 67 1 is 0 Now we have the tools to nish our example f96 b y bm z by y logbx f 1z 091 So if f bw7 then f 1x logbx Property 01 F07quot b gt 0 and b 31 1 loglem z 7 since b bm s0 7 a blogbm since logbx logb s0 7 a Example 04 Usmg these properties 0 log446 6 7109723 23 How do we graph a logarithmic function We could nd pick a series of values7 nd corresponding y values and draw a smooth graph between them This method is inef cient and unsatisfactory Recall that the inverse of a function is a re ection of the function across the line y x So we can take points from the graph f bm7 switch the z and y coordi nates and graph them Example 05 Graph fx 5m and g log5 m the same coordinate system By switching the z and y values found above we can nd points on f 1z Since we know already that f 1x g then this gives us a way to graph WC Property 02 Some Properties 1 The domain of logbx consists of all positive real numbers 07 The range of logbx consists of all real numbers 7007 PW The graphs of all logarithmic functions of the form logbx pass through the point 10 because logbl 0 because b0 1 That is the xintercept is 1 There is no yintercept 9s Ifb gt 1 logbx has a graph that is increasing Ifb lt 1 logbx has a graph that is decreasing 5 The gmph of togbx has a vertical asymptote at x 0 When we want to nd the domain of a more complicated logarithmic function care must be taken togbx is valid for values of x gt 07 so togbx h is valid for values where x h gt 0 Example 06 Find the domain of tog5x 6 is de ned where x 6 gt 0 where x gt 76 Domain of z is 767 There are a number of special logarithms that get used frequently When no base is specified7 it is understood that the logarithm is base 10 The book refers to this as the common logarithmi The logarithmic function with base e is called the natural logarithmi e z 27182818 and we write togex tnxi Example 07 091000 09101000 3 t0g108 toglo108 8 1010953 10t0g1053 53 tne 1 tne7 7 617L300 lnelsm 13x