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Logarithmic Functions

Logarithmic Functions. Lesson 2.5. How to Graph These Numbers?. Consider the vast range of the numbers. 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.

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Logarithmic Functions

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  1. Logarithmic Functions Lesson 2.5

  2. How to Graph These Numbers? • Consider thevast rangeof the numbers

  3. How to Graph These Numbers? • What's wrongwith thispicture?

  4. How to Graph These Numbers? • What's wrongwith thispicture? We need a way toset a scale that fits all the data

  5. How to Graph These Numbers? • The solution:Set the scaleto be the exponentof the distance This is called a logarithmic scale

  6. 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 numberis an exponent

  7. Note: if no base specified, default is base of 10 The Log Function • Try Theselog39 = ? log232 = ? log 0.01 = ?

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

  9. 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

  10. Change of Base Theorem • Create a function for your calculator • Definefunction • Try it • Verify

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

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

  13. 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

  14. Assignment • Lesson 2.5 • Page 121 • Exercises 1 – 79 EOO

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