Transcript Lesson 2.5
Logarithmic Functions Lesson 2.5 How to Graph These Numbers? Distance from the Sun Consider the vast range of the numbers Object Distance (million km) Mercury 58 Venus 108 Earth 149 Mars 228 Jupiter 778 Saturn 1426 Uranus 2869 Neptune 4495 Pluto 5900 Proxima Centauri 4.1E+07 Andromeda Galaxy 2.4E+13 2 How to Graph These Numbers? Distance (million km) 3E+13 2.5E+1 3 2E+13 1.5E+1 3 1E+13 5E+12 tu rn U ra nu s N ep tu ne Pr ox Pl i ut An m a o C dr om ent au ed ri a G al ex y r Sa pi te s Ju M ar rth Ea nu s Ve cu ry 0 M er What's wrong with this picture? 3 How to Graph These Numbers? What's wrong with this picture? Distance (million km) 6000 5000 4000 3000 2000 1000 o C en m ed t au r a G i al ex y Pl ut dr o a im An We need a way to set a scale that fits all the data Pr ox M er cu ry Ve nu s Ea rth M ar s Ju pi te r Sa tu rn U ra n N us ep tu ne 0 4 How to Graph These Numbers? The solution: Set the scale to be the exponent of the distance Distance (million km) 1E+14 1E+12 1E+10 1E+08 1E+06 10000 100 ta y ed a G C en ro m a xi m An d Pr o This is called a logarithmic scale al ex ur i o Pl ut tu ne N ep n nu s U ra ur Sa t pi te r Ju ar s M th Ea r us Ve n M er cu ry 1 5 A New Function Consider the exponential function y = 10x Based on that function, declare a new function x = log10y You should be able to see that these are inverse functions x In general a b log b a x The log of a number is an exponent 6 The Log Function Try These log39 = ? 3 9 so log 3 9 2 2 log232 = ? 25 32 so log 2 32 5 log 0.01 = ? Note: if no base specified, 102 default 0.01 is base of 10 so log 0.01 2 7 Properties of Logarithms Note box on page 105 of text Most used properties log(a b) log a log b a log log a log b b n log a n log a 8 Change of Base Theorem To find the log of a number for a base other than 10 or e … • Use log b x log a x log b a Note new spreadsheet assignment on Blackboard Where b can be any base • Typically 10 or e • Available on calculator 9 Change of Base Theorem Create a function for your calculator • Define function • Try it • Verify 10 Solving Log Equations Use definition of logarithm • Rewrite log equation as an exponential equation 5 log 4 x 2 Result 5 2 4 x x = 32 11 Solving Exponential Equations Use property of logarithms log a n log a or n ln e n n .01 x Consider 5e 9 • Isolate exponential expression • Take ln of both sides • Solve for x 12 Doubling Time What if inflation is at the 5% rate … How long until prices double? 2 P P 1.05 t Strategy • Divide through by P • Take log of both sides • Bring t out as coefficient • Solve for t 13 Assignment Lesson 2.5 Page 121 Exercises 1 – 79 EOO 14