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  • 1. 5.4: Common & Natural Logarithmic Functions © 2008 Roy L. Gover(www.mrgover.com) Learning Goals: •Evaluate common and natural logarithms •Solve logarithmic equations
  • 2. Definition logay x= if and only if y=log base a of x y a x=
  • 3. Important Idea logay x= Logarithmic Form Exponential Form y a x=
  • 4. The logarithmic function is the inverse of the exponential function y x= Important Idea x y a= logay x=
  • 5. Example Write the following logarithmic function in exponential form: 10log 0.01 2= − 5log 25 2= 3 1 log 3 27   = −    6log 6 1=
  • 6. Try This Write the following logarithmic function in exponential form: 2 10 100=10log 100 2= 2 1 5 25 − =5 1 log 2 25   = −   
  • 7. Important Idea In your book and on the calculator, is the same as . If no base is stated, it is understood that the base is 10. 10log x log x
  • 8. Example Without using your calculator, find each value: log1000 log1 log 10 log( 3)−
  • 9. Try This Without using your calculator, find each value: log100,000 log10 3 log 10 log( 7)− 5 1 1/3 undefined
  • 10. Example Solve each equation by using an equivalent statement: log 2x = 10 29x =
  • 11. Try This Solve each equation by using an equivalent statement: log 3x = 10 52x = x=1000 x=1.716
  • 12. Definition A second type of logarithm exists, called the natural logarithm and written ln x, that uses the number e as a base instead of the number 10. The natural logarithm is very useful in science and engineering.
  • 13. Important Idea Like , the number e is a very important number in mathematics. π
  • 14. Important Idea ln x The natural logarithm is a logarithm with the base e is a short way of writing: loge x
  • 15. Definition ln x y= y e x= If and only if ln logex x=
  • 16. Example ln 0.0198 Use a calculator to find the following value to the nearest ten-thousandth:
  • 17. Try This 1 ln 0.32       Use a calculator to find the following value to the nearest ten-thousandth: 1.1394
  • 18. Example Solve each equation by using an equivalent statement: ln 4x = 5x e =
  • 19. Try This Solve each equation by using an equivalent statement: ln 2x = x=7.389 x=2.0798x e =
  • 20. Example Using your calculator, graph the following: 1 lny x= Where does the graph cross the x-axis?
  • 21. Example Using your calculator, graph the following: 1 lny x= Can ln x ever be 0 or negative?
  • 22. Example Using your calculator, graph the following: 1 lny x= What is the domain and range of ln x?
  • 23. Example Using your calculator, graph the following: 1 lny x= How fast does ln x grow? Find the ln 1,000,000.
  • 24. Try This Using your calculator, graph: 1 lny x= 2 ln( 3)y x= − 3 ln( 3) 5y x= − + Describe the differences. How does the domain and range change?
  • 25. Lesson Close A logarithm is an exponent. Illustrate with an example why this is so.