Logarithmic Functions Lesson 2.5

How to Graph These Numbers? Consider the vast range of the numbers Distance from the Sun 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

How to Graph These Numbers? What's wrong with this picture?

How to Graph These Numbers? What's wrong with this picture? We need a way to set a scale that fits all the data

How to Graph These Numbers? The solution: Set the scale to be the exponent of the distance This is called a logarithmic scale

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 In general The log of a number is an exponent

Note: if no base specified, default is base of 10 The Log Function Try These log39 = ? log232 = ? log 0.01 = ? Note: if no base specified, default is base of 10

Properties of Logarithms Note box on page 105 of text Most used properties

Note new spreadsheet assignment on Blackboard Change of Base Theorem To find the log of a number for a base other than 10 or e … Use Where b can be any base Typically 10 or e Available on calculator Note new spreadsheet assignment on Blackboard

Change of Base Theorem Create a function for your calculator Define function Try it Verify

Solving Log Equations Use definition of logarithm Result x = 32 Rewrite log equation as an exponential equation Result x = 32

Solving Exponential Equations Use property of logarithms Consider Isolate exponential expression Take ln of both sides Solve for x

Doubling Time What if inflation is at the 5% rate … How long until prices double? Strategy Divide through by P Take log of both sides Bring t out as coefficient Solve for t

Assignment Lesson 2.5 Page 121 Exercises 1 – 79 EOO