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Exponential and Logarithm functions

Exponential and Logarithm functions. Objectives: • To define exponential and logarithmic functions . • To investigate the properties of exponential and logarithmic functions. • To introduce some applications of exponential and logarithmic functions.

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Exponential and Logarithm functions

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  1. Exponential and Logarithm functions Objectives: • To define exponential and logarithmic functions. • To investigate the properties of exponential and logarithmic functions. • To introduce some applications of exponential and logarithmic functions. • To solve exponential and logarithmic equations.

  2. Exponential Function If is a positive number and is any number, we define the exponential function as: Domain: All real numbers Range: y > 0

  3. Graph of the exponential function

  4. Example x y 0 1 1 3 y 2 9 (0,1) x

  5. properties of the exponential functions

  6. Example Simplify the expression

  7. Example Solve the equation Solution:

  8. The logarithmic Function If is any positive number other than , then the logarithm of to the base denoted by: (a) 1. Domain: (0, ) • Range: (- 3. x-intercept: (1, 0) 4. Continuous on (0, ) 5. Increasing on (0, ) if a> 1

  9. The graph can be obtained by reflecting the graph of across the 45. Note: Logarithmic functions are the inverse of exponential functions. For example if (0, 1) is a point on the graph of an exponential function, then (1, 0) would be the corresponding point on the graph of the inverse logarithmic function.

  10. Rules for base a logarithms

  11. Example write each of the following in terms of simpler logarithms: ) Solution:

  12. )= =] =

  13. =0.5; ==0; ==-2; ==1; =; =

  14. The Natural Logarithms Function The natural logarithmic function is a logarithms function with base e not a. At a = e = 2.7182828…, we get the natural logarithm and denoted by:

  15. Properties of Logarithms For any numbers and the natural logarithm satisfies the following:

  16. The Natural Exponential Function to base e For every real number

  17. Laws of exponents for ex 1-= 2-= 3-= 4-==

  18. Note

  19. Example =3 =5 =0.301

  20. Example In following Solve y in terms of x : Solution: -1 )

  21. 2. ) =

  22. )= = = =+1

  23. Example Use the properties of logarithms to simplify the following expressions: • -

  24. Solution = 3. - = = =

  25. Example By using logarithms and exponentials properties as needed, solve the following for x:

  26. Solution = =

  27. 2. =

  28. 3. 4.

  29. Example Suppose that a cup of soup cooled from 90°C to 60°C after 10 min in a room whose temperature was 20°C. Use Newton’s law of cooling to answer the following questions. a. How much longer would it take the soup to cool to 35°C? b. Instead of being left to stand in the room, the cup of 90°C soup is put in a freezer whose temperature is Howlong will it take the soup to cool from 90°C to 35°C?

  30. Solution =20, =60 = it will take

  31. = - =

  32. Example If E is the energy released, measured in joules, during an earthquake then the magnitude of the earthquake is given by, How much energy will be released in an earthquake with a magnitude of 5.9?

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