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Exponential and Logarithm functions

Learn about exponential and logarithmic functions, their properties, applications, and how to solve equations related to them. Understand the relationship between exponential and logarithmic functions and explore practical examples.

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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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